Try This Number Theory Problem

Number Theory Level 1

An integer is a multiple of 9 if the sum of its digits is a multiple of 9. In the above example, we know that $135$ is a multiple of 9 because $1 + 3 + 5 = 9$ is a multiple of 9.

Which of the following numbers is a multiple of 9?

12345678 12345 123456 1234567

3 solutions

Calvin Lin Staff
Jul 10, 2014

Let's find the sum of the digits.
$1 + 2 + 3 + 4 + 5 = 15$ which is not a multiple of 9, so $12345$ is not a multiple of 9. :(
$1 + 2 + 3 + 4 + 5 +6 = 21$ which is not a multiple of 9, so $1234567$ is not a multiple of 9. :(
$1 + 2 + 3 + 4 + 5 +6 + 7 = 28$ which is not a multiple of 9, so $1234567$ is not a multiple of 9. :(
$1 + 2 + 3 + 4 + 5 +6 + 7 + 8 = 36$ which is a multiple of 9, so $12345678$ is a multiple of 9!

I think there's a typo (12345 is not a multiple of 9) and (1234567 is not of multiple of 9) (the bolded words should be added). I'm unable to edit your solution despite being a moderator.

Victor Loh - 6 years, 11 months ago

Edited. Thank you! I'm not sure how I missed that out :(

Calvin Lin Staff - 6 years, 11 months ago

9! is a factorial. So change it to 9.

Munem Shahriar - 3 years, 9 months ago

@Munem Shahriar – This is a case where proper punctuation helps you convey what you mean. Because it is not written as "9!.", I mean "9 exclamation mark" instead of "9 factorial (no ending punctuation)".

Calvin Lin Staff - 3 years, 9 months ago

Munem Shahriar
Sep 15, 2017

$\huge 12345678 \div 9 = 1371742$

$\huge 1371742 \times 9 = \boxed{12345678}$

Dang Anh Tu
Jul 11, 2014

Thank you, Calvin. We can use this method to find a multiple of 3, too!

Yes, the numbers whose digits add up to a multiple of 3, will be a multiple of 3 $:)$

Ashish Menon - 5 years, 3 months ago

Try This Number Theory Problem

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