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# Given two positive integers A and B such that A > B, what is the remai

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Manager
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Given two positive integers A and B such that A > B, what is the remai  [#permalink]

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23 Mar 2017, 02:37
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Difficulty:

65% (hard)

Question Stats:

52% (01:20) correct 48% (01:26) wrong based on 44 sessions

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Given two positive integers A and B such that A > B, what is the remainder when the square of B is subtracted from the square of A and then divided by 15?

(1) When the sum of A and B is divided by 5, the remainder is 1

(2) When B is subtracted from A and then divided by 3, the remainder is 1.
Senior Manager
Joined: 13 Oct 2016
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Re: Given two positive integers A and B such that A > B, what is the remai  [#permalink]

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23 Mar 2017, 12:49
2
Spartan85 wrote:
Given two positive integers A and B such that A > B, what is the remainder when the square of B is subtracted from the square of A and then divided by 15?

(1) When the sum of A and B is divided by 5, the remainder is 1

(2) When B is subtracted from A and then divided by 3, the remainder is 1.

Hi

(1) When the sum of A and B is divided by 5, the remainder is 1

A + B = 5x + 1. ------> A + B = 6, 11, 16 ... Putting this into remainders we can have (0 + 1), (1 + 0), (2 + 4), (4 + 2).

We are getting different values for A^2 - B^2 and remainders as a result. Insufficient.

(2) When B is subtracted from A and then divided by 3, the remainder is 1.

A - B = 3y + 1. Same logic as in previous case. Insufficient.

(1)&(2) Combining two:

$$A + B = 6, 11, 16, 21, 26, 31, 36 ....$$
$$A - B = 1, 4, 7, 10, 13, 16, 19, 22 ...$$

A + B = 11, A - B = 1 ----> A = 6, B = 5 ------->$$\frac{A^2 - B^2}{15} = \frac{36 - 25}{15} = \frac{16}{15}$$ ----> remainder 1.

A + B = 16, A - B = 4 ----> A = 10, B = 6 -------> $$\frac{A^2 - B^2}{15} = \frac{100 - 36}{15} = \frac{64}{15}$$ ----> remainder 4.

Insufficient

Another way to look at this:

$$A^2 - B^2 = (A - B)*(A + B) = (5x + 1)(3y+1) = 15xy + 5x + 3y + 1.$$

We can ingnore $$15xy$$ which brings remainder $$0$$. $$\frac{5x + 3y + 1}{15}$$ will give us diferent remainders for different values of $$x$$ and $$y$$. Insufficient.

Re: Given two positive integers A and B such that A > B, what is the remai   [#permalink] 23 Mar 2017, 12:49
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