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§ 4-3 Greatest Common Factor and Least Common Multiple Greatest Common Divisor Given two whole numbers, how can we define the greatest common divisor (GCD)? Greatest Common Divisor Given two whole numbers, how can we define the greatest common divisor (GCD)? Definition The greatest common divisor of two non-zero whole numbers is the largest whole number that divides both numbers. Greatest Common Divisor Given two whole numbers, how can we define the greatest common divisor (GCD)? Definition The greatest common divisor of two non-zero whole numbers is the largest whole number that divides both numbers. Notation gcd(a, b) (a, b) Finding Greatest Common Divisors We will discuss 3 methods for finding greatest common divisors: 1 Set Intersections Finding Greatest Common Divisors We will discuss 3 methods for finding greatest common divisors: 1 Set Intersections 2 Prime Factorization Finding Greatest Common Divisors We will discuss 3 methods for finding greatest common divisors: 1 Set Intersections 2 Prime Factorization 3 the Euclidean Algorithm The Set Intersection Method Example Find the gcd of 100 and 36. The Set Intersection Method Example Find the gcd of 100 and 36. To use this method, we list all of the factors each. This method is good when neither whole number has a lot of factors and when the numbers are such that we can quickly find all of the factors. The Set Intersection Method Example Find the gcd of 100 and 36. To use this method, we list all of the factors each. This method is good when neither whole number has a lot of factors and when the numbers are such that we can quickly find all of the factors. Factors of 100 The Set Intersection Method Example Find the gcd of 100 and 36. To use this method, we list all of the factors each. This method is good when neither whole number has a lot of factors and when the numbers are such that we can quickly find all of the factors. Factors of 100 { 1, 2, 4, 5, 10, 20, 25, 50, 100 } The Set Intersection Method Example Find the gcd of 100 and 36. To use this method, we list all of the factors each. This method is good when neither whole number has a lot of factors and when the numbers are such that we can quickly find all of the factors. Factors of 100 { 1, 2, 4, 5, 10, 20, 25, 50, 100 } Factors of 36 The Set Intersection Method Example Find the gcd of 100 and 36. To use this method, we list all of the factors each. This method is good when neither whole number has a lot of factors and when the numbers are such that we can quickly find all of the factors. Factors of 100 { 1, 2, 4, 5, 10, 20, 25, 50, 100 } Factors of 36 { 1, 2, 3, 4, 6, 9, 12, 18, 36} The Set Intersection Method Example Find the gcd of 100 and 36. To use this method, we list all of the factors each. This method is good when neither whole number has a lot of factors and when the numbers are such that we can quickly find all of the factors. Factors of 100 { 1, 2, 4, 5, 10, 20, 25, 50, 100 } Factors of 36 { 1, 2, 3, 4, 6, 9, 12, 18, 36} The Set Intersection Method Example Find the gcd of 100 and 36. To use this method, we list all of the factors each. This method is good when neither whole number has a lot of factors and when the numbers are such that we can quickly find all of the factors. Factors of 100 { 1, 2, 4, 5, 10, 20, 25, 50, 100 } Factors of 36 { 1, 2, 3, 4, 6, 9, 12, 18, 36} Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. 2100 = Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. 2100 = 22 · 3· 52 · 7 Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Model We would use this model with whole numbers that we may not be able to easily list all of the factors but when the prime factorization is attainable. Example Find the gcd of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 gcd(2100, 6370) = 2 · 5 · 7 = 70 The Euclidean Algorithm What did the division algorithm say? The Euclidean Algorithm What did the division algorithm say? The Division Algorithm For whole numbers a and b with a ≥ b, there exists unique whole numbers q and r such that a = bq + r where 0 ≤ r < b The Euclidean Algorithm What did the division algorithm say? The Division Algorithm For whole numbers a and b with a ≥ b, there exists unique whole numbers q and r such that a = bq + r where 0 ≤ r < b We can use this to find greatest common divisors. Remember that d|a, d|b ⇒ d|(a + b) The Euclidean Algorithm The Euclidean Algorithm If a and b are whole numbers such that a = bq + r as described in the Division Algorithm, then gcd(a, b) = gcd(b, r) The Euclidean Algorithm The Euclidean Algorithm If a and b are whole numbers such that a = bq + r as described in the Division Algorithm, then gcd(a, b) = gcd(b, r) This is best used when we have large numbers that are not easy to find the prime factorization for or when there is a large difference between a and b. The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 4830 = 3786(1) + 1044 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 4830 = 3786(1) + 1044 3786 = 1044(3) + 654 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 4830 = 3786(1) + 1044 3786 = 1044(3) + 654 1044 = 654(1) + 390 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 4830 = 3786(1) + 1044 3786 = 1044(3) + 654 1044 = 654(1) + 390 654 = 390(1) + 264 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 4830 = 3786(1) + 1044 3786 = 1044(3) + 654 1044 = 654(1) + 390 654 = 390(1) + 264 390 = 264(1) + 126 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 4830 = 3786(1) + 1044 3786 = 1044(3) + 654 1044 = 654(1) + 390 654 = 390(1) + 264 390 = 264(1) + 126 264 = 126(2) + 12 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 4830 = 3786(1) + 1044 3786 = 1044(3) + 654 1044 = 654(1) + 390 654 = 390(1) + 264 390 = 264(1) + 126 264 = 126(2) + 12 126 = 12(10) + 6 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 4830 = 3786(1) + 1044 3786 = 1044(3) + 654 1044 = 654(1) + 390 654 = 390(1) + 264 390 = 264(1) + 126 264 = 126(2) + 12 126 = 12(10) + 6 12 = 6(2) + 0 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 4830 = 3786(1) + 1044 3786 = 1044(3) + 654 1044 = 654(1) + 390 654 = 390(1) + 264 390 = 264(1) + 126 264 = 126(2) + 12 126 = 12(10) + 6 12 = 6(2) + 0 gcd(32766, 4830) = gcd(4830, 3786) The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 gcd(32766, 4830) = gcd(4830, 3786) 4830 = 3786(1) + 1044 gcd(4830, 3786) = gcd(3786, 1044) 3786 = 1044(3) + 654 1044 = 654(1) + 390 654 = 390(1) + 264 390 = 264(1) + 126 264 = 126(2) + 12 126 = 12(10) + 6 12 = 6(2) + 0 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 gcd(32766, 4830) = gcd(4830, 3786) 4830 = 3786(1) + 1044 gcd(4830, 3786) = gcd(3786, 1044) 3786 = 1044(3) + 654 gcd(3786, 1044) = gcd(1044, 654) 1044 = 654(1) + 390 654 = 390(1) + 264 390 = 264(1) + 126 264 = 126(2) + 12 126 = 12(10) + 6 12 = 6(2) + 0 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 gcd(32766, 4830) = gcd(4830, 3786) 4830 = 3786(1) + 1044 gcd(4830, 3786) = gcd(3786, 1044) 3786 = 1044(3) + 654 gcd(3786, 1044) = gcd(1044, 654) 1044 = 654(1) + 390 gcd(1044, 654) = gcd(654, 390) 654 = 390(1) + 264 390 = 264(1) + 126 264 = 126(2) + 12 126 = 12(10) + 6 12 = 6(2) + 0 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 gcd(32766, 4830) = gcd(4830, 3786) 4830 = 3786(1) + 1044 gcd(4830, 3786) = gcd(3786, 1044) 3786 = 1044(3) + 654 gcd(3786, 1044) = gcd(1044, 654) 1044 = 654(1) + 390 gcd(1044, 654) = gcd(654, 390) 654 = 390(1) + 264 gcd(654, 390) = gcd(390, 264) 390 = 264(1) + 126 264 = 126(2) + 12 126 = 12(10) + 6 12 = 6(2) + 0 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 gcd(32766, 4830) = gcd(4830, 3786) 4830 = 3786(1) + 1044 gcd(4830, 3786) = gcd(3786, 1044) 3786 = 1044(3) + 654 gcd(3786, 1044) = gcd(1044, 654) 1044 = 654(1) + 390 gcd(1044, 654) = gcd(654, 390) 654 = 390(1) + 264 gcd(654, 390) = gcd(390, 264) 390 = 264(1) + 126 gcd(390, 264) = gcd(264, 126) 264 = 126(2) + 12 126 = 12(10) + 6 12 = 6(2) + 0 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 gcd(32766, 4830) = gcd(4830, 3786) 4830 = 3786(1) + 1044 gcd(4830, 3786) = gcd(3786, 1044) 3786 = 1044(3) + 654 gcd(3786, 1044) = gcd(1044, 654) 1044 = 654(1) + 390 gcd(1044, 654) = gcd(654, 390) 654 = 390(1) + 264 gcd(654, 390) = gcd(390, 264) 390 = 264(1) + 126 gcd(390, 264) = gcd(264, 126) 264 = 126(2) + 12 gcd(264, 126) = gcd(126, 12) 126 = 12(10) + 6 12 = 6(2) + 0 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 gcd(32766, 4830) = gcd(4830, 3786) 4830 = 3786(1) + 1044 gcd(4830, 3786) = gcd(3786, 1044) 3786 = 1044(3) + 654 gcd(3786, 1044) = gcd(1044, 654) 1044 = 654(1) + 390 gcd(1044, 654) = gcd(654, 390) 654 = 390(1) + 264 gcd(654, 390) = gcd(390, 264) 390 = 264(1) + 126 gcd(390, 264) = gcd(264, 126) 264 = 126(2) + 12 gcd(264, 126) = gcd(126, 12) 126 = 12(10) + 6 gcd(126, 12) = gcd(12, 6) 12 = 6(2) + 0 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 gcd(32766, 4830) = gcd(4830, 3786) 4830 = 3786(1) + 1044 gcd(4830, 3786) = gcd(3786, 1044) 3786 = 1044(3) + 654 gcd(3786, 1044) = gcd(1044, 654) 1044 = 654(1) + 390 gcd(1044, 654) = gcd(654, 390) 654 = 390(1) + 264 gcd(654, 390) = gcd(390, 264) 390 = 264(1) + 126 gcd(390, 264) = gcd(264, 126) 264 = 126(2) + 12 gcd(264, 126) = gcd(126, 12) 126 = 12(10) + 6 gcd(126, 12) = gcd(12, 6) 12 = 6(2) + 0 gcd(12, 6) = 6 The Euclidean Algorithm Example Find the greatest common divisor of 32766 and 4830. 32766 = 4830(6) + 3786 gcd(32766, 4830) = gcd(4830, 3786) 4830 = 3786(1) + 1044 gcd(4830, 3786) = gcd(3786, 1044) 3786 = 1044(3) + 654 gcd(3786, 1044) = gcd(1044, 654) 1044 = 654(1) + 390 gcd(1044, 654) = gcd(654, 390) 654 = 390(1) + 264 gcd(654, 390) = gcd(390, 264) 390 = 264(1) + 126 gcd(390, 264) = gcd(264, 126) 264 = 126(2) + 12 gcd(264, 126) = gcd(126, 12) 126 = 12(10) + 6 gcd(126, 12) = gcd(12, 6) 12 = 6(2) + 0 gcd(12, 6) = 6 So, gcd(32766, 4830) = 6. A Theorem Remember that if d|a and d|b then d|(a − b). We can use this when finding greatest common divisors. A Theorem Remember that if d|a and d|b then d|(a − b). We can use this when finding greatest common divisors. Theorem For whole numbers a and b, gcd(a, b) = gcd(a − b, b). A Theorem Remember that if d|a and d|b then d|(a − b). We can use this when finding greatest common divisors. Theorem For whole numbers a and b, gcd(a, b) = gcd(a − b, b). We can extend this as well. Theorem For whole numbers a and b and positive whole number k, gcd(a, b) = gcd(a − kb, b), A Theorem Remember that if d|a and d|b then d|(a − b). We can use this when finding greatest common divisors. Theorem For whole numbers a and b, gcd(a, b) = gcd(a − b, b). We can extend this as well. Theorem For whole numbers a and b and positive whole number k, gcd(a, b) = gcd(a − kb, b), This theorem is the reason that the Euclidean Algorithm really works. Least Common Multiples Who can define the least common multiple of two whole numbers a and b? Least Common Multiples Who can define the least common multiple of two whole numbers a and b? Least Common Multiple (LCM) For whole numbers a and b, the least common multiple of a and b is te smallest whole number with a and b as factors. Least Common Multiples Who can define the least common multiple of two whole numbers a and b? Least Common Multiple (LCM) For whole numbers a and b, the least common multiple of a and b is te smallest whole number with a and b as factors. Notation: 1 lcm(a, b) Least Common Multiples Who can define the least common multiple of two whole numbers a and b? Least Common Multiple (LCM) For whole numbers a and b, the least common multiple of a and b is te smallest whole number with a and b as factors. Notation: 1 lcm(a, b) 2 [a, b] Number Line Model At first introduction, we could use number lines to explicitly illustrate to students how to find lcms. Number Line Model At first introduction, we could use number lines to explicitly illustrate to students how to find lcms. Example Find the least common multiple of 2 and 5. 0 1 2 3 4 5 6 7 8 9 10 Number Line Model At first introduction, we could use number lines to explicitly illustrate to students how to find lcms. Example Find the least common multiple of 2 and 5. 0 1 2 3 4 5 6 7 8 9 10 Number Line Model At first introduction, we could use number lines to explicitly illustrate to students how to find lcms. Example Find the least common multiple of 2 and 5. 0 1 2 3 4 5 6 7 8 9 10 Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 Prime Factorization Method The number line method would be very tedious, however, with larger numbers, and we wouldn’t want to use an illustrated method with older students. But we can always go back to prime factorizations. Example Find the LCM of 2100 and 6370. 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 lcm(2100, 6370) = 22 · 3 · 52 · 72 · 13 LCM-GCD Method LCM-GCD Method For whole numbers a and b, a · b = lcm(a, b) · gcd(a, b) Why? 2100 = LCM-GCD Method LCM-GCD Method For whole numbers a and b, a · b = lcm(a, b) · gcd(a, b) Why? 2100 = 22 · 3· 52 · 7 LCM-GCD Method LCM-GCD Method For whole numbers a and b, a · b = lcm(a, b) · gcd(a, b) Why? 2100 = 22 · 3· 52 · 7 6370 = LCM-GCD Method LCM-GCD Method For whole numbers a and b, a · b = lcm(a, b) · gcd(a, b) Why? 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 LCM-GCD Method LCM-GCD Method For whole numbers a and b, a · b = lcm(a, b) · gcd(a, b) Why? 2100 = 22 · 3· 52 · 7 6370 = 2· 5· 72 · 13 2100 · 6370 = 23 · 3 · 53 · 73 · 13 LCM-GCD Method So how do we use it? It is sometimes easier to find the factors than the multiples ... Example Find the LCM of 3420 and 4290. LCM-GCD Method So how do we use it? It is sometimes easier to find the factors than the multiples ... Example Find the LCM of 3420 and 4290. We again begin with prime factorization. LCM-GCD Method So how do we use it? It is sometimes easier to find the factors than the multiples ... Example Find the LCM of 3420 and 4290. We again begin with prime factorization. 3420 = 22 · 32 · 5 · 19 LCM-GCD Method So how do we use it? It is sometimes easier to find the factors than the multiples ... Example Find the LCM of 3420 and 4290. We again begin with prime factorization. 3420 = 22 · 32 · 5 · 19 4290 = 2 · 3 · 5 · 11 · 13 LCM-GCD Method So how do we use it? It is sometimes easier to find the factors than the multiples ... Example Find the LCM of 3420 and 4290. We again begin with prime factorization. 3420 = 22 · 32 · 5 · 19 4290 = 2 · 3 · 5 · 11 · 13 Then, the product: 3420 · 4290 = 14671800 LCM-GCD Method So how do we use it? It is sometimes easier to find the factors than the multiples ... Example Find the LCM of 3420 and 4290. We again begin with prime factorization. 3420 = 22 · 32 · 5 · 19 4290 = 2 · 3 · 5 · 11 · 13 Then, the product: 3420 · 4290 = 14671800 Note: This is a multiple of our two numbers, just not necessarily the smallest one. When would the product be the LCM? LCM-GCD Method Now that we have the product, we need to find the GCD. LCM-GCD Method Now that we have the product, we need to find the GCD. 3420 = 22 · 32 · 5 · 19 4290 = 2 · 3 · 5 · 11 · 13 LCM-GCD Method Now that we have the product, we need to find the GCD. 3420 = 22 · 32 · 5 · 19 4290 = 2 · 3 · 5 · 11 · 13 gcd(3420, 4290) = 2 · 3 · 5 = 30 LCM-GCD Method Now that we have the product, we need to find the GCD. 3420 = 22 · 32 · 5 · 19 4290 = 2 · 3 · 5 · 11 · 13 gcd(3420, 4290) = 2 · 3 · 5 = 30 So, we can now find the LCM. 14671800 = 30 · lcm(3420, 4290) LCM-GCD Method Now that we have the product, we need to find the GCD. 3420 = 22 · 32 · 5 · 19 4290 = 2 · 3 · 5 · 11 · 13 gcd(3420, 4290) = 2 · 3 · 5 = 30 So, we can now find the LCM. 14671800 = 30 · lcm(3420, 4290) 14671800 lcm(3420, 4290) = 30 LCM-GCD Method Now that we have the product, we need to find the GCD. 3420 = 22 · 32 · 5 · 19 4290 = 2 · 3 · 5 · 11 · 13 gcd(3420, 4290) = 2 · 3 · 5 = 30 So, we can now find the LCM. 14671800 = 30 · lcm(3420, 4290) 14671800 lcm(3420, 4290) = 30 lcm(3420, 4290) = 489060 LCM of More Than 2 Whole Numbers We do not have to find the LCM of 3 or more numbers separately and then combine. There are ways we can do it in one shot. One way is using a common factor grid. Example Find the LCM of 12, 18 and 30. LCM of More Than 2 Whole Numbers We do not have to find the LCM of 3 or more numbers separately and then combine. There are ways we can do it in one shot. One way is using a common factor grid. Example Find the LCM of 12, 18 and 30. 12 18 30 LCM of More Than 2 Whole Numbers We do not have to find the LCM of 3 or more numbers separately and then combine. There are ways we can do it in one shot. One way is using a common factor grid. Example Find the LCM of 12, 18 and 30. 2 12 18 30 LCM of More Than 2 Whole Numbers We do not have to find the LCM of 3 or more numbers separately and then combine. There are ways we can do it in one shot. One way is using a common factor grid. Example Find the LCM of 12, 18 and 30. 2 12 18 30 6 9 15 LCM of More Than 2 Whole Numbers We do not have to find the LCM of 3 or more numbers separately and then combine. There are ways we can do it in one shot. One way is using a common factor grid. Example Find the LCM of 12, 18 and 30. 2 12 18 30 3 6 9 15 LCM of More Than 2 Whole Numbers We do not have to find the LCM of 3 or more numbers separately and then combine. There are ways we can do it in one shot. One way is using a common factor grid. Example Find the LCM of 12, 18 and 30. 2 12 18 30 3 6 9 15 2 3 5 LCM of More Than 2 Whole Numbers We do not have to find the LCM of 3 or more numbers separately and then combine. There are ways we can do it in one shot. One way is using a common factor grid. Example Find the LCM of 12, 18 and 30. 2 12 18 30 3 6 9 15 2 3 5 LCM of More Than 2 Whole Numbers We do not have to find the LCM of 3 or more numbers separately and then combine. There are ways we can do it in one shot. One way is using a common factor grid. Example Find the LCM of 12, 18 and 30. 2 12 18 30 3 6 9 15 2 3 5 lcm(12, 18, 30) = 22 · 32 · 5 = 180