Question

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

Answer 1

I hope anyone can help

Answer 2

9997 x = 1000x - 3x
x clearly has to be an odd no.
well i assume no such x exists
as for some odd no. say x=2n+1,,we have
1000(2n+1) -3(2n+1)
1994n + 1000 -3
now 1994n will always be an even no.
also 1994 + 1000 will still let it be an even no.
1994n can never be a no. with all digits either odd or even
same with 1000+ 1994n
-3 will only make changes with ones digit(or sometimes tens digit)..
so according to me,,it has no soln..
i may be wrong..hmmn..

Answer 3

nic qn

Answer 4

ok do any have an answer cuz there is an answer for it :)

Answer 5

Well, since 9997=10000-3, and you're looking for an n such that 9997n has all odd digits, then 10000n-3n must have all odd digits.

If you look at 10000n, only the first number could possibly be odd. If you subtract anything with an even number in any of the digits (for example, 455), then you will be left with something with an even digit. Therefore, anything we subtract from 10000n must have all odd digits. We are subtracting 3n from 10000n, so 3n must have all odd digits.

Answer 6

Hm, @shubhamsrg you might be right about it having no solution.

Any time you have 10000n where n is odd, then you will end up with an odd number in the first digit. When you subtract 3n (which will be odd), you will end up with an even first digit in your final answer.

Any time you have 10000n where n is even, then you will end up with an even number in the first digit, but 3n will be even.

Answer 7

so what is the smallest multiple ??

Answer 8

hmmn..

Answer 9

actually,,there should be a solution..
10001 gives an ans.. @jabberwock
dont know if any no. less than that is there..

Answer 10

Hm...right you are.

Answer 11

Oh, I see where I messed up.

Answer 12

http://answers.yahoo.com/question/index?qid=20081021020708AAD9cJz

Answer 13

There's a solution online already.

Answer 14

but 759775 isnt a multiple of 9997 !

Answer 15

@shubhamsrg and jabberwock If 9997k is odd, then k must be odd. 9997k = 10000k − 3k
= 10000(k − 1) + (10000 − 3k). Since k > 1, 10000 − 3k < 10000. Therefore, if 10000 −
3k is positive, 10000(k − 1) has an even digit in the ten-thousands place. Therefore, we
must have 10000 − 3k < 0, so k > 3333. Testing the first possibility, we find that
(9997)(3335) = 33339995 has no even digits.

Answer 16

@shubhamsrg and jabberwock really thank u for your help :)

Answer 17

It should have said 749775, which is 75*9997.