Best way to divide a number by a large prime number

When you rewrite the division as 10150/29, you aren't simplifying - you're making your life more complicated. Probably the most useful thing to learn if you want to do pen-and-paper math quickly is this: cancel *first*, multiply *last*. By canceling, you get smaller numbers, and life becomes much easier. So here we have:

406*100/116 = (203)(50)/29

Now, if the answer is an integer, the 29 must cancel, and it doesn't cancel with 50, so we need to divide 203 by 29. There are all kinds of ways to do that, but if 29k = 203, k must be less than 10, and by considering that the product of the units digits must end in 3, you can see that k = 7, pretty quickly, so (203)(50)/29 = 7*50 = 350.

The way that I do these problems is to break down larger numbers into their prime factors (makes that cancelling step a bit more obvious/more difficult to make mistakes).

406 = 2 * 7 * 29
100 = 2 * 2 * 5 *5
116 = 2 * 2 * 29

2^3 * 5^2 * 7 *29 / 2^2 * 29 = 2 * 5^2 * 7 or written out...

2*2*2*5*5*7*29
-----------------
2*2*29

14 * 25 = 350

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