We have two responses for you

Hi Peter.

Let b = the integer quotient of N divided by d excluding the remainder. That means bd + 7 = N. Another way of writing this is

Now consider 5N. That equals 5bd + 35. When this is divided by d, we have 5b + 35/d, so the fractional part 35/d gives rise to the remainder of 10.

35/25 = 1 remainder 10. So d is 25.

Check:
N/25 = b remainder 7. 5N/25 = 5b remainder 35 = 5b + (1 remainder 10). The check works.

Cheers,
Stephen La Rocque

You have N = kd+7 and 5N = pd+10 for integers k & d. Thus 5kd+35 = pd + 10 and d must divide 25 (why?) Thus d = 1, 5 or 25. Dividing N by d leaves a remainder of 7 so that d couldn't be 1 (why?). Dividing 5N by d leaves a remainder of 10 so that d couldn't be 5 (why?). Thus ...?

Penny