Question 1110462
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We have to have {{{(9b+7)/(b+7)}}} equal to a whole number; and presumably the value(s) of b we are looking for are integers greater than or equal to 2.<br>
{{{(9b+7)/(b+7) = ((9b+63)-56)/(b+7) = 9 - 56/(b+7)}}}<br>
Since 9 is a whole number, {{{56/(b+7)}}} has to be a whole number also.<br>
So we know b has to be a whole number greater than or equal to 2; and we know 56 has to be divisible by (b+7).  The only divisors of 56 that are 9 or greater are 14, 28, and 56.<br>
Since those divisors are the values of (b+7), the three bases for which (b+7) divides into (9b+7) without any remainder are 7, 21, and 49.<br>
Check:
b=7: (9b+7)(b+7) = 70/14 = 5
b=21: (9b+7)/b+7) = 196/28 = 7
b=49: (9b+7)/b+7) = 448/56 = 8