After the Division of a Number Successively by 3, 4, and 7 GMAT Problem Solving

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?

1. 80
2. 75
3. 41
4. 53
5. 60

Answer:

Solution with Explanation:
Approach Solution (1):

Given that when positive integer x is divided successively by 3, 4, and 7, the remainders are 2, 1, and 4, respectively.

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 (let’s say A) is divided by 7, the remainder is 4

Dividend: Divisor * Quotient + Remainder

A: Dividend
7: Divisor
t: quotient (Assume)
4: Remainder
A = 7 * t + 4 = 7t + 4

When an integer (let’s say B) is divided by 4, gives A as quotient and 1 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

Correct Option: D

Approach Solution (2):

We will 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.ie 84x + 53

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

Correct Option: D

“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?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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