The remainder when the positive integer m is divided by 7 is x. The re

Question

The remainder when the positive integer m is divided by 7 is x. The remainder when m is divided by 14 is x + 7. Which one of the following could m equal?

(A) 45

(B) 53

(C) 72

(D) 85

(E) 100

EgmatQuantExpert · Accepted Answer

Solution

Given:  
 • The remainder when positive integer m is divided by 7 is x
• The remainder when m is divided by 14 is (x + 7)

To find:  
 • Out of the given options, which one can be equal to m

Approach and Working:   
When m is divided by 7, remainder is x. If we assume the quotient to be a, then
 • The integer m = 7a + x

Similarly, when m is divided by 14, remainder is (x + 7). If we assume the quotient to be b, then
 • The integer m = 14b + (x + 7) = 14b + x + 7 = 7 (2b + 1) + x

Comparing both the expressions of m, 
 • 7a + x = 7 (2b + 1) + x
We can write a = 2b + 1
Or, b =  \frac{(a – 1)}{2}

As b is an integer, a must be an odd number

Out of the given options, only 53 gives an odd quotient, when divided by 7.

Hence, the correct answer is option B.

Answer: B

Note: Verifying the options to get the answer is definitely another way of solving this problem

warrior1991 · Answer

When divided by 14 the remainder is 7 plus some x. Therefore remainder will always >7 when divided by 14.

Only B satisfies the condition.
53 when divided by 14 leaves remainder 11.therefore x=4 and 53 when divided by 7 also leaves remainder 4.

IMO B.

anuj04 · Answer

When m is divided by 14 the remainder is 7+x. 
=>remainder has to be more than 7
Easiest way is to check with the options when divided by 14.
(A) 45 - remainder is 3-WRONG
(B) 53 - Remainder is 11- Keep it for time being.
(C) 72 - remainder is 2-WRONG
(D) 85 - remainder is 1-WRONG
(E) 100 - remainder is 2-WRONG

Only option B satisfies.

B is the answer.

sumitkrocks · Answer

m=7k+x ......(1) Also, m=14k'+7+x .........(2)
from 2nd expression m=7(2k'+1)+x

We can infer there is constraint put by the 2nd condition, quotient of the expression shall be an odd number when divided by 7 .

Lets examine options

(A) 45 =7*6+3 Incorrect 
 
(B) 53 = 7*7 +4 Correct

(C) 72 =7*10+2 Incorrect

(D) 85 =7*12+1 Incorrect

(E) 100=7*14+2 Incorrect

KSBGC · Answer

For this question Back solving is the finest way. Note that when m is divided by 7 remainder is x but when x is divided by 14 the remainder becomes x+7.

A) 45/7= (6*7)+3
but 45/14=(14*3)+3

So A is eliminated as remainder is same in both cases.

B) 53/7=(7*7)+4

53/14=(14*3)+11. Look at the remainder here. Remainder is 11. we know x=4 when m is divided by 7. Now we have remainder x+7=4+7=11

Thus, Option B meets the condition. Bis the correct answer.

ScottTargetTestPrep · Answer

We see that 14 = 7(2). So when a number is divided by 7 and by 14, it either produces the same remainder (e.g. 15/7 = 2 R 1 and 15/14 = 1 R 1) or produces a different remainder (e.g. 13/7 = 1 R 6 or 13/14 = 0 R 13). We can see that in the latter example, the remainder from dividing by 14 is 7 more than the remainder from dividing by 7. In other words, when a number is divided by 7, the quotient must be the odd number, as shown in the latter example (if the quotient is even, as shown in the former example, then the two remainders will be the same).
So let’s check each answer choice (remember we are looking for an odd quotient).

A) 45

45/7 = 6 R 3

Quotient is not odd. So A is not the answer.

B) 53

53/7 = 7 R 4

Quotient is odd. So B is the answer.

Answer: B

nigina93 · Answer

Its B and here is why,
to be it seemed easy to jump directly to answer choices and work back. Start either from B or D so you know which direction to move if B/D dont work. I started with B. if m is 53, then dividing by 7 gives remainder of 4, if 53/14, remainder is 11 (x(4)=7). mark B and move on

auradediligodo · Answer

Hi everyone,

The approach to find the answer is the following.

Divide each number by 7 and then by 14.
The remainder found when the number is divided by 7 is X.
So the remainder to be found when the same number is divided by 14 is X+7.

As we can see option B satisfies the condition and hence it is the correct choice.

Option B

BrushMyQuant · Answer

Theory: Dividend = Divisor*Quotient + Remainder

Given that the remainder when the positive integer m is divided by 7 is x and the remainder when m is divided by 14 is x + 7. And we need to find which of the following could be the value of m

Let's solve the problem using    Substitution

We will take each option choice and find out the remainder with 7 and 14 and see which one has remainder by 14, 7 greater than the remainder by 7.

(A) 45 
45 when divided by 7 gives 3 remainder
45 when divided by 14 gives 3 remainder
Clearly, Remainder by 14 is NOT 7 greater than Remainder by 7 =>   NOT POSSIBLE

(B) 53 
53 when divided by 7 gives 4 remainder
53 when divided by 14 gives 11 remainder
Clearly, Remainder by 14 IS 7 greater than Remainder by 7 =>   POSSIBLE  
In Test, we don't need to solve further, but I am solving to complete the solution.

(C) 72 
72 when divided by 7 gives 2 remainder
72 when divided by 14 gives 2 remainder
Clearly, Remainder by 14 is NOT 7 greater than Remainder by 7 =>   NOT POSSIBLE

(D) 85 
85 when divided by 7 gives 1 remainder
85 when divided by 14 gives 1 remainder
Clearly, Remainder by 14 is NOT 7 greater than Remainder by 7 =>   NOT POSSIBLE

(E) 100 
100 when divided by 7 gives 2 remainder
45 when divided by 14 gives 2 remainder
Clearly, Remainder by 14 is NOT 7 greater than Remainder by 7 =>   NOT POSSIBLE

So,  Answer will be B 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

HarshavardhanR · Answer

-> x>=0. Because it is the remainder when m is divided by 7 (remainder cannot be negative). 
-> When m is divided by 14, the remainder is x + 7. Given x>=0, x+7 has to be >=7.

We can run through the choices to see if we can eliminate any choices. For instance, we can already see that a choice like 45 (A) should not be correct. 45 when divided by 14 gives a remainder of 3.

A) 45 when divided by 14 -> remainder 3. Does not fit x+7 above.
B) 
-> 53 when divided by 14 -> remainder 11. 
-> 53 when divided by 7 -> remainder 4
-> This fits "x+7". x = 4. 4 + 7 = 11, the remainder when the same 53 is divided by 14.  B is our answer .

We do not even have to check C,D, and E because B clearly could be m.

But for the sake of completeness ->

C) 72 when divided by 14 -> remainder 2. Does not fit x+7 above. 
D) 85 when divided by 14 -> remainder 1. Does not fit x+7 above. 
E) 100 when divided by 14 -> remainder 2. Does not fit x+7 above.

---
Harsha

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