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 Author: questioner [ Mon Mar 26, 2012 6:24 am ] Post subject: GMAT Number Theory Positive integer n is divisible by 5.What is the remainder when dividing 4(n + 1)(n + 8) by 5?A. 0B. 1C. 2D. 3E. 4(C) Let’s simplify 4(n + 1)(n + 8) = 4(n² + 9n + 8) = 4n² + 36n + 32. Or we can put it in this way:4(n + 1)(n + 8) = (4n² + 36n + 30) + 2.n is divisible by 5 and 30 is also divisible by 5 so (4n² + 36n + 30) is divisible by 5. Therefore 2 is the remainder when dividing 4(n + 1)(n + 8) by 5. The correct answer is C.An alternative way to solve this is to plug in numbers for n and you would be able to arrive at the answer.----------I tried plugging in 5 and 10 for N and every time got a different remainder. Not sure what am I doing wrong...

 Author: Gennadiy [ Mon Mar 26, 2012 6:52 am ] Post subject: Re: GMAT Number Theory Quote:I tried plugging in 5 and 10 for N and every time got a different remainder. Not sure what am I doing wrong...Apparently you had a mistake in the calculations.4(5 + 1)(5 + 8) = 312312 is 310 + 2. 310 is divisible by 10 and therefore by 5. So the remainder is 2.4(10 + 1)(10 + 8) = 792792 is 790 + 2. 790 is divisible by 10 and therefore by 5. So the remainder is 2.Here is a way to do the calculations without a calculator:4(5 + 1)(5 + 8) = 4 × 6 × 13 = 4 × 78 = 3124(10 + 1)(10 + 8) = 4 × 11 × 18 = 11 × 72 = 792What shortcuts are used here?1. Multiplication by 11 is quite simple. It is the same as multiply by 10 + the number we multiply.72 × 11 = 72 × 10 + 72 × 1 = 720 + 72 = 792.So any other multiplication by 11 will be the same:77 × 11 = 770 + 77 = 847273 × 11 = 2730 + 273 = 3003This trick can also be used for multiplication by 12, 13, 103 etc.72 × 12 = 720 + 72 × 2 = 720 + 144 = 86472 × 103 = 7200 + 72 × 3 = 7200 + 216 = 7416 2. Generally, it's easier to multiply by a digit, so instead of4 × 6 × 13 = 24 × 13we can do4 × (6 × 13) = 4 × 78Note, that calculating 24 × 13 using the previous method is also quite fast here. 24 × 13 = 240 + 24 × 3 = 240 + 72 = 312 3. Multiplication by a digit we do mentally as in long multiplication method:4 × 78 =First, we multiply 4 × 8 = 32. So the unit digit will be 2Then, we multiply 4 × 7 = 28 and add 3 that we got from 32 in the previous step. 28 + 3 = 31The result is 312.These are the shortcuts, but in any case you should use the method most convenient to you. And always pick the easiest numbers to plug in first.

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