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A positive two-digit integer ab is a multiple of a-b. How many such integers are there?

 Apr 6, 2021
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If $a-b=1$ there are $10, 21, 32, ..., 98=9$ integers.

If $a-b=2$ then we have $20, 42, 64, 86=4$ integers.

If $a-b=3$, then we have $30, 63, 96=3$ integers.

If $a-b=4$, then we have $40, 84=2$ integers.

If $a-b=5$, then we have $50=1$ integer.

If $a-b=6$, then we have $60$.

If $a-b=7$, we have $70$, if $a-b=8$ we have $80$, if $a-b=9$ we have $90$.

 

So it's $9+4+3+2+1+1+1+1+1=\boxed{23}$. 

 Apr 6, 2021

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