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

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Darknight123
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After the division of a number successively by 3, 4 and 7 [#permalink] New post  22 Aug 2021, 03:44
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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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Re: After the division of a number successively by 3, 4 and 7 [#permalink] New post  22 Aug 2021, 06:25
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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.
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Re: After the division of a number successively by 3, 4 and 7 [#permalink] New post  24 Aug 2021, 02:04
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Number = 3(4(7x+4)+1)+2

= 84x+53

D
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Re: After the division of a number successively by 3, 4 and 7 [#permalink] New post  06 Nov 2022, 07:44
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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


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Re: After the division of a number successively by 3, 4 and 7 [#permalink]
06 Nov 2022, 07:44
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