Inspired by Shreya R . (9)

Number Theory Level 2

Find the smallest positive multiple of 9 with all digits even and different. (no repetition).

This problem is inspired from problem named 9 by Shreya R This question is a part of this set .


The answer is 468.

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Nihar Mahajan by Nihar Mahajan , 20, India

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5 solutions

Danish Ahmed
Jan 25, 2015

The sum of digits of every multiple of 9 9 is itself a multiple of 9 9 .

Every 2 2 -digited multiple of 9 9 contains an odd digit.

The sum of digits of all 3 3 -digited multiples of 9 9 are : 9 9 , 18 18 , 27 27 .

If the sum of digits is 9 9 or 27 27 , it contains atleat one odd digit.

18 = 4 + 6 + 8 18 = 4 + 6 + 8 . So the smallest 3 3 -digited multiple of 9 9 with all digits even and different is 468 468 .

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Noel Lo
Jun 8, 2015

A number with all even digits HAS to be even. If it is also divisible by 9, it is a multiple of 18.

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Moderator note:

How does that help?

It reduces the number of possibilities I need to consider. By trial and error, the smallest multiple of 18 which satisfies the given condition is 468.

Noel Lo - 6 years ago

The sum of digits has to be 18. It has to be made up of 3 digits (the biggest even digits are 6 and 8, 6+8=14<18). The only way to achieve a sum of 18 with three different even digits is 8+6+4. The smallest number that can be created with these numbers and thus the smallest multiple of 9 satisfying the given conditions is 468.

Eric Scholz - 2 years, 3 months ago
Refath Bari
Jul 14, 2016

There we go! :) Hope my explanation below is clear enough!:

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Luis Ortiz
Jan 9, 2016

If the sum of digits in a number is a multiple of 9 then the number is divisible by 9.

n + w + x = 9z

Since n,w,x must be even, then 9z = 2k. Smallest possiblity is 18.

n + w + x = 18

Since n,w,x must be even and different numbers, we find the combination (by trial and error) 4,8,6. The order must be 468 to make it the smallest possible number that solves the question.

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Ramiel To-ong
Sep 7, 2015

same analysis.

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