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Theory Given that an integer "n" when divided by an integer "a" gives "r" as remainder then the integer "n" can be written as n = ak + r where k is a constant integer.
Example1. What is the remainder when B is divided by 6 if B is a positive integer? (1) When B is divided by 18, the remainder is 3 (2) When B is divided by 12, the remainder is 9
Sol STAT1 : When B is divided by 18, the remainder is 3 So, we can write B as B = 18k + 3 Now, to check the remainder when B is divided by 6, we essentially need to check the remainder when 18k + 3 is divided by 6 18k goes with 6 so the remainder will 3 So, its sufficient
STAT2 : When B is divided by 12, the remainder is 9 So, we can write B as B = 12k + 9 Now, to check the remainder when B is divided by 6, we essentially need to check the remainder when 12k + 9 is divided by 6 12k goes with 6 so the remainder will be the same as the remainder for 9 divided by 6 which is 3 So, remainder is 3 So, its sufficient Answer will be D Link to the problem: http://gmatclub.com/forum/what-is-the-r ... 42343.html
Example2: What is the remainder when positive integer t is divided by 5? (1) When t is divided by 4, the remainder is 1 (2) When t is divided by 3, the remainder is 1
Sol: STAT1: When t is divided by 4, the remainder is 1 t = 4k +1 possible values of t are 1,5,9,13 Clearly we cannot find a unique remainder when t is divided by 5 as in some cases(t=1) we are getting the remainder as 1 and in some(t=5) we are getting the remainder as 0. So, INSUFFICIENT
STAT2: When t is divided by 3, the remainder is 1 t = 3s + 1 possible values of t are 1,4,7,10,13,16,19 Clearly we cannot find a unique remainder when t is divided by 5 as in some cases(t=1) we are getting the remainder as 1 and in some(t=10) we are getting the remainder as 0. So, INSUFFICIENT
taking both together now there are two approaches 1. write the values of t from stat1 and then from stat2 and then take the common values from STAT1 t = 1,5,9,13,17,21,25,29,33 from STAT2 t = 1,4,7,10,13,16,19,22,25,28,31,34 common values are t = 1,13,25,
2. equate t = 4k+1 to t=3s+1 we have 4k + 1 = 3s+1 k = 3s/4 since, k is an integer so only those values of s which are multiple of 4 will satisfy both STAT1 and STAT2 so, common values are given by t = 3s + 1 where s is multiple of 4 so t = 1,13,25 (for s=0,4,8 respectively)
Clearly we cannot find a unique remainder when t is divided by 5 as in some cases(t=1) we are getting the remainder as 1 and in some(t=10) we are getting the remainder as 0. So, INSUFFICIENT So, answer will be E
Example3:If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15 ? (1) The remainder when p + n is divided by 5 is 1. (2) The remainder when p - n is divided by 3 is 1
Sol: STAT1 : The remainder when p + n is divided by 5 is 1. p+n = 5k + 1 but we cannot say anything about p^2 - n^2 just from this information. So, INSUFFICIENT
STAT2 : The remainder when p - n is divided by 3 is 1 p-n = 3s + 1 but we cannot say anything about p^2 - n^2 just from this information. So, INSUFFICIENT
Taking both together p^2 - n^2 = (p+n) * (p-n) = (5k + 1) * (3s + 1) = 15ks + 5k + 3s + 1 The remainder of the above expression by 15 is same as the remainder of 5k + 3s + 1 with 15 as 15ks will go with 15. But we cannot say anything about the remainder as its value will change with the values of k and s. So INSUFFICIENT Hence answer will be E
Example 4: If n is a positive integer and r is the remainder when 4 + 7n is divided by 3, what is the value of r? (1) n+1 is divisible by 3 (2) n>20.
Sol: r is the remainder when 4 + 7n is divided by 3 7n + 4 can we written as 6n + n + 3+ 1 = 3(2n+1) + n +1 remainder of 7n+4 by 3 will be same as remainder of 3(2n+1) + n +1 by 3 3*(2n+1) will go by 3 so the remainder will be the same as the remainder of (n+1) by 3
STAT1: n+1 is divisible by 3 n+1 = 3k (where k is an integer) n+1 will give 0 as the remainder when divided by 3 so, 7n+4 will also give 0 as the remainder when its divided by 3 (as its remainder is same as the remainder for (n+1) when divided by 3) => r =0 So, SUFFICIENT
STAT2 n>20. we cannot do anything by this information as there are many values of n so, INSUFFICIENT.
Example 5: If x is an integer, is x between 27 and 54? (1) The remainder when x is divided by 7 is 2. (2) The remainder when x is divided by 3 is 2.
Sol: STAT1: The remainder when x is divided by 7 is 2. x = 7k + 2 Possible values of x are 2,9,16,...,51,... we cannot say anything about the values of x so, INSUFFICIENT
STAT2: The remainder when x is divided by 3 is 2. x = 3s + 2 Possible values of x are 2,5,8,11,...,53,... we cannot say anything about the values of x so, INSUFFICIENT
Taking both together now there are two approaches 1. write the values of t from stat1 and then from stat2 and then take the common values from STAT1 x = 2,9,16,23,30,37,44,51,58,...,65,... from STAT2 x = 2,5,8,...,23,...,44,...,59,65,... common values are x = 2,23,44,65,...
2. equate x = 7k+2 to x=3s+2 we have 7k + 2 = 3s+2 k = 3s/7 since, k is an integer so only those values of s which are multiple of 7 will satisfy both STAT1 and STAT2 so, common values are given by x = 3s + 2 where s is multiple of 7 so x = 2,23,44,65 (for s=0,7,14,21 respectively)
Clearly there are values of x which are between 27 and 54 (i.e. 44) and those which are not (2,23,65) So, both together also INSUFFICIENT So, answer will be E
Re: How To Solve: Remainder Problems
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