Multiple of 9997 with no even digits

By Maria Jose Leon-Sotelo Esteban (T3819) on Wednesday, January 17, 2001 - 07:24 am :

What is the smallest integer multiple of 9997,other than 9997 itself,which contains only odd digits?

Maria Jose.

By James Lingard (Jchl2) on Wednesday, January 17, 2001 - 03:08 pm :

3335 \times 9997 = 33339995

. This is the smallest such multiple.

Let's consider $9997 n$ . First note that $n$ must be odd. Then, because $9997 = 10000 - 3$ , if $n \leq 3333$ then $3 n < 10000$ and so $9997 n$ looks like

$[n - 1] [10000 - 3 n]$

that is, the first few digits of $9997 n$ are the number (n-1) and the remaining digits are the number (10000-3n) (with leading zeros if necessary). But since $n$ is odd, the last digit of $n - 1$ is even, so that's no good.

So $n > 3333$ , and trying the first odd values of $n$ greater than 3333 (i.e. 3335) works.

James.