GMAT math remainder Questions... Too Tough

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I have collected these problems on remainder. This type of problem is frequently asked in DS.
Answers are also given. Please dont mind any typo error.

1.If r is the remainder when the positive integer n is divided by 7, what is the value of r

1. when n is divided by 21, the remainder is an odd number
2. when n is divided by 28, the remainder is 3

The possible reminders can be 1,2,3,4,5 and 6. We have the pinpoint the exact remainder from this 6 numbers.

St 1: when n is divided by 21 ( 7 and 3) the remainder is an odd number.
But it cannot be 7, 3 or 9 . Hence the possibilities are : 1 and 5.
Hence there can be two remainders ,1 and 5, when divided by 7.
NOT SUFFICIENT

St 2: when n is divided by 28 the remainder is 3.
As 7 is a factor of 28, the remainder when divided by 7 will be 3
SUFFICIENT


2 If n and m are positive integers, what is the remainder when 3^(4n + 2 + m) is divided by 10 ?
(1) n = 2
(2) m = 1

The Concept tested here is cycles of powers of 3.

The cycles of powers of 3 are : 3,9,7,1

St I) n = 2. This makes 3^(4*2 +2 + m) = 3^(10+m). we do not know m and hence cannot figure out the unit digit.

St II) m=1 . This makes 3^(4*n +2 + 1).
4n can be 4,8,12,16...
3^(4*n +2 + 1) will be 3^7,3^11, 3^15,3^19 ..... in each case the unit digit will be 7. SUFF
Hence B


3.If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.

(2) p is the sum of the squares of two positive integers.


st1. take multiples of 8....divide them by 4...remainder =1 in each case...

st2. p is odd ,since p is square of 2 integers...one will be even and other odd....now when we divide any even square by 4 v ll gt 0 remainder..and when divide odd square vll get 1 as remainder......so intoatal remainder=1
Ans : D


4.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.

Ans: E

st1) p+n=6,11,16....insuff.
st2) p-n=4,7,10....insuff...

multiply these two to get p^2-n^2.....multiplying any ttwo values from the above results in different remainder......

also can be done thru equation....p+n=5a+1..and so on


5.What is the remainder when the positive integer x is divided by 3 ?

(1) When x is divided by 6, the remainder is 2.

(2) When x is divided by 15, the remainder is 2.

Easy one , answer D

st 1...multiple of 6 will also be multiple of 3 so remainder wil be same as 2.

st2) multiple of 15 will also be multiple of 3....so the no.that gives remaindr 2 when divided by 15 also gives 2 as the remainder when divided by 3...


6.What is the remainder when the positive integer n is divided by 6 ?

(1) n is a multiple of 5.

(2) n is a multiple of 12.

Easy one. Answer B

st 1) multiples of 5=5,10,15....all gives differnt remainders with 6

st2)n is divided by 12...so it will be divided by 6...remainder=0


7If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7 ?

(1) y = 6

(2) z = 3

We need to know all the variables. We cannot get that from both the statements. Hence the answer is E.

8.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

Answer A
st1... n+1 divisible by 3..so n=2,5,8,11......
this gives 4+7n=18,39,60....remainder 0 in each case......
st2) insufficient ....n can have any value


9.If n is a positive integer and r is the remainder when (n - 1)(n + 1) is divided by 24, what is the value of r ?

(1) n is not divisible by 2.

(2) n is not divisible by 3.

ST 1- if n is not divisible by 2, then n is odd, so both (n - 1) and (n + 1) are even. moreover, since every other even number is a multiple of 4, one of those two factors is a multiple of 4. so the product (n - 1)(n + 1) contains one multiple of 2 and one multiple of 4, so it contains at least 2 x 2 x 2 = three 2's in its prime factorization.
But this is not sufficient, because it can be (n-1)*(n+1) can be 2*4 where remainder is 8. it can be 4*6 where the remainder is 0.

ST 2- if n is not divisible by 3, then exactly one of (n - 1) and (n + 1) is divisible by 3, because every third integer is divisible by 3. therefore, the product (n - 1)(n + 1) contains a 3 in its prime factorization.
Just like st 1 this is not sufficient

the overall prime factorization of (n - 1)(n + 1) contains three 2's and a 3.
therefore, it is a multiple of 24.
sufficient

Answer C

유사 관계대명사 용법

보기A번

He bought sth for three times the price as I paid for it. (X)

유사관계대명사 as를 쓰려면, 선행사에 as/such/the same 와 같은 수식어가 있어야 쓸 수 있어요.

=> He bought sth for three times the price as I paid for it. (O)

A is sold for three times the price as the maker charges for B. (X)

A is sold for three times the price than the maker charges for B. (X)

A is sold for three times the price that the maker charges for B. (O) 배수사의 용법!

= A is sold for three times as high price as the maker charges for B. (O) 유사관계대명사 용법

= A is sold for three times higher price than the maker charges for B. (O) 유사관계대명사 용법

A is sold for three times the price of what the maker charges for B. (X)

A is sold for three times the price at which the maker charges for B. (X)

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<참고> 유사관계대명사 - as/ than/ but

 

1. as (선행사에 as/such/the same 이 있을 때)

He will buy such a car as speeds like a bullet. (주격) 유사함을 의미하는 such 때문에 that을 쓰면 안되고 as를 써야 함.

He will buy such a car as he saw yesterday. (목적격)

 

She is absent, as is often her case. (주격) 그녀가 결석인데, 그건 그녀에게 종종 있는 경우다

As is often the case, she is absent. (문두에 놓여서 관용구로 쓰임) 종종 있는 일이지만, 그녀는 결석이다.

 

* 외워두세요

As is often the case with ~ (~에게 흔히 있는 일이지만)

 

* 참고

He bought the same car as I bought. ==> 같은 종류의 서로 다른 두 개

He bought the same car that I bought. (X) ==> 동일한 한 개. 이 문장은 illogical 하므로 틀린 문장.

 

Tom loves the same girl as John loves. => 두 명의 similar girls

Tom loves the same girl that John love. => 한 명의 동일한 girl

 

2. than (선행사에 비교급이 있을 때)

She has more money than is necessary. (주격)

She has more money than she needs. (목적격)

 

3. but (선행사에 부정어가 있을 때 - no/not/rarely/little/less/few/scarcely/seldom...)

There is no boy but knows her name. (but 은 부정어가 들어있다) 그녀의 이름을 모르는 소년은 아무도 없다.

(= There is no boy who does not know her name.)

(= Every boy knows her name.)