Download Category 3 – Number Theory – Meet #2 – Practice #1

Category 3 – Number Theory – Meet #2 – Practice #1
1) If the GCF is 12, then the third number must be divisible by 12, but not have anything else in
common with 48(12x4) or 72(12x6). The multiples of 12 between 110 and 150 are 120(12x10),
132(12x11), and 144(12x12). Both 120 and 144 have an addition factor of 2 with both 48 and 72, so
neither of those can be the third number. 132 is 12x11 and 11 is not a factor of 48 or 72, so 132
must be the third number!!
2) It is always the case that the GCF of two numbers times the LCM of the same two numbers is equal
to the product of the two numbers. [GCF(x,y)][LCM(x,y)] = xy, so the product of x and y in this
case is 12x144 = 1728.
3) The factors of 240 that are divisible by 6 are {6, 12, 24, 30, 48, 60, 120} and
The factors of 462 that are divisible by 6 are {6, 42, 66}
Since 120 and 66 have a GCF of 6 and those are each the largest factor of their respective numbers,
the largest value of x + y = 120 + 66 = 186
4) 1680 = 24 ⋅ 31 ⋅ 51 ⋅ 71 so the sum of the exponents = 4 + 1 + 1 + 1 = 7
5) The GCF( 2 3 ⋅ 31 ⋅ 5 0 ⋅ 7 2 ⋅ 112 , 2 5 ⋅ 30 ⋅ 51 ⋅ 71 ⋅ 111 ) = 23 ⋅ 30 ⋅ 50 ⋅ 71 ⋅111 = 8 ⋅1 ⋅1 ⋅ 7 ⋅11 = 616
6) If the GCF(x, y) = 48 then x can be written as 48a and y can be written as 48b. If the GCF(y, z) = 72
then y can be written as 72c and z can be written as 72d.
So we are looking for GCF(48a, 48b, 72c, 72d) = 24. Basically just the GCF(48,72) = 24
7) The GCF(a, b) = 28 and the LCM(a, b) = 420. If a = 84, what is b?
Since it is always true that [GCF (a, b) ][ LCM (a, b)] = ab , ab = 28 ⋅ 420 and
b=
28 ⋅ 420 28 ⋅ 420 1 ⋅ 420
=
=
= 140
a
84
3
8) Two numbers have a product of 770. What is their LCM?
Since 770 = 770 = 2 ⋅ 5 ⋅ 7 ⋅11 , the two numbers would have to split that prime factorization. Since all
of the factors are to the first power, the two numbers must be relatively prime.
For example (10 and 77) or (14 and 55). The LCM then is going to also be 770 either way!!!
9) Mike has a piece of plywood that measures 5.5 ft by 6 ft. He wants to cut it into squares all the same
size and as big as possible. If x is the number of square inches in the largest possible squares he
could make, and y is the number of those squares, what is the positive difference of x and y?
5.5 ft = 66 inches, and 6 ft = 72 inches, he can cut it into squares with side length GCF(66,72) = 6
So the squares are each 6 on a side, so the area of each is 6x6=36 which is x. He can cut the 5.5 ft
into eleven 6 inch parts, and the 72 inches into twelve 6 inch parts. Leaving a total of 11x12 = 132
squares which is y. The positive difference between x and y is 132 – 36 = 96