After the division of a number successively by 3, 4 and 7

Question

After the division of a number successively by 3, 4 and 7, the remainders obtained are 2,1 and 4 respectively. What will be the remainder if 84 divides the same number?

(A) 80
(B) 75
(C) 41
(D) 53
(E) 60

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Bhu750 · Accepted Answer

We need to work in the reverse order,

Remainder after division by 7 is 4,
Number before division by 7, 7x+4 
Number before division by 4, 4(7x+4)+1, i.e. 28x+17
Number before division by 3, 3(28x+17)+2, i.e. 84x+53

Now 84x is a multiple of 84 and the remainder is 0 and for 53/84 the remainder is 53.

rvgmat12 · Answer

Number = 3(4(7x+4)+1)+2

= 84x+53

D

BrushMyQuant · Answer

Given that When positive integer x is divided successively by 3, 4, and 7, the remainders are 2, 1, and 4, respectively and we need to find the remainder when x is divided by 84 ?

We need to solve this problem backwards  
 - Something divided by 7 and gives 1 as remainder.
- Take that number as quotient for division by 4 and remainder as 1.
- Take the new number formed as the quotient for division by 3 and remainder as 2 to get the value of x.
- Then divide x by 84 to check the remainder.

When an integer (lets say A) is divided by 7, the remainder is 4

Theory: Dividend = Divisor*Quotient + Remainder

A -> Dividend
7 -> Divisor
t -> Quotient (Assume)
4 -> Remainder
=> A = 7*t + 4 = 7t + 4

When an integer (Lets say "B") is divided by 4, gives A as quotient and 1 as remainder

=> B = 4A + 1 = 4*(7t + 4) + 1 = 28t + 16 + 1 = 28t + 17

When x is divided by 3, gives B as quotient and 2 as remainder

=> x = 3B + 2 = 3*(28t + 17) + 2 = 84t + 51 + 2 = 84t + 53

=> x when divided by 84 gives 53 remainder

So,  Answer will be D 
Hope it helps!

Watch the following video to learn the Basics of Remainders

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

Shuvojoti · Answer

Why did we approach this math backward? Why can't we go forward by dividing gradually with 3,4 and then 7?

Oppenheimer1945 · Answer

N=3(4(7z+4)+1) +2=3(28z+17)+2=84z+53

ankita3101 · Answer

Yes, that's my question too!

Oppenheimer1945 · Answer

That's more direct. Backtrack adds further complexity.

bumpbot · Answer

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After the division of a number successively by 3, 4 and 7

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