Best way to divide a number by a large prime number

Question

Hi there,

I just came across a question in the Kaplan 800 book that just took me too long (q12 - word problems)

In order to solve this problem, I need to divide 406 * 100 by 116. This can be simplified to 10150 / 29

My problem here is that 29 is a pretty large prime number and it just takes me too long to come up with the solution by using long division. What method would you recommend to solve 10150 / 29 faster?

Thanks!
Steve

Aleehsgonji · Answer

406*100/116 = 203*100/58
Next time when the above expression is divided by 2, do not divide 100 but divide 203
= 101.5*100/29
Looking we can say 101.5 is divisible by 29 by 3 times.
So subtract 101.5 - 87 = 14.5
14.5 is half of 29
So 101.5*100/29 = 3.5*100 = 350

ntgmat · Answer

Am not sure of the options given for the question, but you can approximate the calculation and then eliminate options.
10150/29 ~ 10150/30 = 338 (approx).

since we have increased the denominator for approximation, the actual answer would be greater than 338. However, this approach might be rendered useless in case the options are close in value (which ideally should not be the case).

Cheers!!

IanStewart · Answer

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.

cereal255 · Answer

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

bumpbot · Answer

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Best way to divide a number by a large prime number

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