## Outline Mathematics

Number Theory

# Smallest multiple of 9 with no odd digits

Here's a problem to tackle:

Find the smallest multiple of 9 with no odd digits.

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

Find the smallest multiple of 9 with no odd digits.

The number we are after has all digits even; at the very least its last digit must be even. Thus the sought number ought be a multiple of 18,12,16,18,24,28. There are just five,two,three,four,five,six 2-digit multiples of 18: 18 36 54 72 90,18 32 50 68 96,18 36 54 72 90. Each has one odd digit.

The 3-digit numbers that start with 1 are those that exceed 99,99,100,101,200 but are below 200,99,100,101,200. Evidently, their first,first,second,third,last digit is odd. So, we have to look furtther up. If we are lucky, there is a multiple of 18 in the 200s with all digits even.

The sum of the digits of such a number is an even multiple of 9. The sum is then divisible by 18. For numbers below 300, the sum is bound to be 18. Excluding the first digit, the sum is 16,9,12,16,18. The only 2-digit number with even digits that add up to 16 is 88,79,86,88. Thus the answer to our problem is 288,88,188,288.

Not very difficult, but still better and more enlightening than mere enumeration of the multiples.

### References

- T. Gardiner,
*More Mathematical Challenges*, Cambridge University Press, 2003, p. 110.

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Copyright © 1996-2018 Alexander Bogomolny