When using factor trees to find prime factorizations of numbers it is important to be able to find factors quickly. Divisibility tests allow us to quickly find factors of numbers. Knowing divisibility tests saves us from having to try dividing by each possible factor to see whether or not it works. You probably already know a few divisibility tests. Here are the most commonly used ones.
A whole number is divisible by 2 if and only if its last digit is 0, 2, 4, 6, or 8. Justification for a three-digit numeral.
A whole number is divisible by 5 if and only if its last digit is 0 or 5. Justification for a three-digit numeral.
A whole number is divisible by 10 if and only if its last digit is 0. Justification for a three-digit numeral.
Example:
Using these tests we can see that 123,456 is divisible by 2 because its last digit is 6. We can also see that 123,456 is not divisible by 5 or 10 because its last digit is not 0 or 5.
A whole number is divisible by 4 if and only if its last two digits are divisible by 4. Justification for a three-digit numeral.
Example:
This test says that to check to see if 123,456 is divisible by 4, we only need to check if 56 is divisible by 4. Since 56 = 14 · 4, we know that 56 is divisible by 4; therefore, 123,456 is also divisible by 4.
A whole number is divisible by 3 if and only if the sum of its digits is divisible by 3. Justification for a three-digit numeral.
A whole number is divisible by 9 if and only if the sum of its digits is divisible by 9. Justification for a three-digit numeral.
Example:
To use this test to check if 123,456 is divisible by 3 or 9, we take the digits of 123,456 and find there sum. We get 1 + 2 + 3 + 4 + 5 + 6 = 21. Since 21 = 7 · 3, we see that the sum of the digits is divisible by 3 and so we also know that 123,456 is divisible by 3.
We also see that since 21 is the sum of the digits, and 21 is not divisible by 9, we know that the number 123,456 is also not divisible by 9.
A whole number is divisible by 6 if and only if it is divisible by both 2 and 3.
This follows directly from what we have studied about prime factorizations. We know that the prime factorization of 6 is 2 · 3. If the prime factorization of a whole number includes the primes 2 and 3, then we know that number has 6 for a factor. And if the number has 6 for a factor, it must be divisible by 6.
Example: We have already shown that the number 123,456 is divisible by both 2 and by 3, so according to this rule the number 123,456 must also be divisible by 6.
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