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Same Count, Same Sum (Posted on 2015-01-06) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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spoiler 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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