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Number Theory – Divisibility
Def.: If a and b are whole numbers and b does not equal zero, and there is a q such that a =
bq, then b divides a, b is a factor of a, and a is a multiple of b. If b divides a and b is less
than a, then b is called a proper factor or a proper divisor of a.
Example: Multiples of 10 are 10, 20, 30, 40, … Factors or divisors of 10 would be 1, 2, 5,
and 10. Proper factors of 10 would be 1, 2, 3, and 5.
Prime numbers are number with exactly two different or unique factors. The smallest
positive prime number is 2. The primes from 1 to 100 can be determined using a hundreds
chart and the process known as the Sieve of Erotosthenes. There is an infinite number of
primes.
The number 1 is not a prime number by this definition because it only has one unique factor
– itself. We call 1 the identity number.
Composite numbers are numbers with more than 2 unique factors.
Fundamental theorem of arithmetic – every natural number >1 is either prime or a product
of primes in one, and only one, way apart from order.
This unique product of primes is called a prime factorization of the number.
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Example: 24  2  3 . This prime factorization may be found using a factor tree, or a
repeated division process.
Prime factor test To test for prime factors of a number n, you only need to check for prime
factors p where p <
n . Example: show that 439 is prime.
Perfect numbers: numbers for which the sum of all the proper factors equal the number.
Deficient numbers: numbers for which the sum of all the proper factors is less than the
number.
Abundant numbers: numbers for which the sum of all the proper factors is greater than the
number.
Teaching activity: Have students list the numbers from 1 to 20. They should then list all of
the proper factors for each of those numbers and determine if the number is perfect, deficient
or abundant.
Germain primes: Let p be a prime number. If 2p + 1 is also a prime number, then p is called
a germain prime. Example: 3 is a germain prime since 2(3) + 1 = 7 is also a prime.
Goldbach Conjecture – every even number greater than 4 can be written as the sum of two
(not necessarily different) primes.
Palindromes – numbers that read the same forwards and backwards.
Divisibility tests
Divisibility test for 2 - the number ends in 0, 2, 4, 6, or 8. (In other words the number is
even.)
Divisibility test for 5 - the number ends in 0 or 5.
Divisibility test for 10 - the number ends in 0. Note: if two and five are factors, so is 10.
Divisibility test for 3 - the sum of the digits of the number is divisible by 3.
Example: Is 924 divisible by 3? 9 + 2 + 4 = 15 and 15 is divisible by 3, therefore 924 is
divisible by 3.
Example: Is 1313 divisible by 3? 1 + 3 + 1 + 3 = 8, 8 is not divisible by 3, therefore 1313
is not divisible by 3.
Divisibility test for 4 - the last two digits of the number are divisible by 4.
Example: Is 986, 216 divisible by 4? The last two digits for the numeral 16, 16 is
divisible by 4, therefore 986,216 is divisible by 4.
Example: Is 123,496 divisible? Note: 96 is 40 + 40 + 16, since all are divisible by 4, 96
is divisible by 4.
Divisibility test for 6 - number is divisible by 2 and 3 (Passes both divisibility tests)
Example: Is 954 divisible by 6? It is divisible by 2, since it ends in 4. It is divisible by 3
since 9 + 5 + 4 = 18 and 18 is divisible by 3. Therefore 954 is divisible by 6.
Divisibility test for 9 - the sum of the digits of the number is divisible by 9.
Example: Is 12, 345, 678 divisible by 9? 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 and 36 is
divisible by 9, therefore 12,345,678 is divisible by 9.
Divisibility test for 11 – the difference of the sums of the digits in the even and odd positions
in the number is divisible by 11.
Divisibility test for 7, 11, and 13 – Break up the number n into a series of three-digits
numbers determined by the three-digit groups starting from the right of n. Add the evennumbered positioned groups and the odd-numbered positioned groups. Subtract these two
sums. If the difference is divisible by 7, then the number n is divisible by 7. Similarly for 11
and 13.