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When positive integer x is divided successively by 3, 4, and 7, the

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Bunuel
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When positive integer x is divided successively by 3, 4, and 7, the [#permalink] New post   28 Aug 2022, 04:18
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When positive integer x is divided successively by 3, 4, and 7, the remainders are 2, 1, and 4, respectively. What is the remainder when x is divided by 84 ?

A. 83
B. 80
C. 75
D. 53
E. 42


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SaquibHGMATWhiz
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Re: When positive integer x is divided successively by 3, 4, and 7, the [#permalink] New post   28 Aug 2022, 04:49
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It is given that, x is divided successively by 3, 4, and 7, and the remainders are 2, 1, and 4.

    We need to understand what is the meaning of successive division. Here, x is divided by 3 and we get a quotient (say Q) and a remainder of 2. We pick the quotient (Q) and then again divide Q by 4 to get another quotient (Q') and a remainder of 1. When we divide this quotient (Q') by 7, we get a quotient of Q'' and a remainder of 4.

We need to start from the end here.


    When we divide this quotient (Q') by 7, we get a quotient of Q'' and a remainder of 4.
      Q' = 7Q'' + 4 ... (1)

    When we divide this quotient (Q) by 4, we get a quotient of Q' and a remainder of 1.
      Q = 4Q' + 1 ... (2)

Thus combining (1) and (2) we have, Q = 4(7Q'' + 4) + 1 ... (3)

    x is divided by 3 and we get a quotient (say Q) and a remainder of 2.
      x = 3Q + 2 .... (4)

Substitution value of Q from (3) to (4) we have,
x = 3[4(7Q'' + 4) + 1] + 2 = 3 [28Q'' + 16 + 1] + 2 = 84Q'' + 53

So when we divide this number by 84 the remainder is 53.

Answer: D
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Re: When positive integer x is divided successively by 3, 4, and 7, the [#permalink] New post   06 Nov 2022, 03:46
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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: When positive integer x is divided successively by 3, 4, and 7, the [#permalink] New post   04 Feb 2024, 23:47
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Bunuel wrote:
When positive integer x is divided successively by 3, 4, and 7, the remainders are 2, 1, and 4, respectively. What is the remainder when x is divided by 84 ?

A. 83
B. 80
C. 75
D. 53
E. 42


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Successive division can be easily handled without any variables by simply starting from the last division step.
Upon division by 7, remainder is 4 which means the smallest such number is 4.
Upon division by 4, the remainder is 1 and quotient is 4 (from previous step) which means the smallest such number is 4*4 + 1 = 17.
Upon division by 3, the remainder is 2 and quotient is 17 (from previous step) which means the smallest such number is 3*17 + 2 = 53.

Hence smallest value of x is 53. When we divide it by 84, the remainder is 53.

Answer (D).

Here is a video discussing Division: https://youtu.be/A5abKfUBFSc
 
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Re: When positive integer x is divided successively by 3, 4, and 7, the [#permalink]
04 Feb 2024, 23:47
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