perplexus.info :: Just Math : Same Count, Same Sum

Same Count, Same Sum (Posted on 2015-01-06)

M and N are two distinct positive integers such that:

(i) M and N has the same number of digits, and:
(ii) Each of the sum of the digits of M and N is equal to 32.

Can M be a multiple of N?

If so, give an example.
If not, prove it.

*** Neither M nor N can contain any leading zero.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Comment 1 of 1

We have M=a*N and (M-N)=0 mod9 since M and N have the same digital root.

Set M=N+9b and substitute to get (a-1)*N = 9b. N is not divisible by 9 since its digital root is 32 so (a-1) is divisible by 9. Since M and N have the same number of digits, a<10. But a<>1 since M<>N. So there are no solutions.

Posted by xdog on 2015-01-06 14:21:57

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