Question: When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k + n is a multiple of 35?

(A) 3
(B) 4
(C) 12
(D) 32
(E) 35

Answer: B
Solution and Explanation:

Approach Solution 1:

Given that a positive integer n is divided by 5 and the remainder is 1. Also when n is divided by 7, the remainder is 3. It has asked to find out the smallest integer k such that k+n will be a multiple of 35.
If you divide a positive integer n by 5, the remainder is 1: n=5q+1, where q is the quotient, and the results are: 1, 6, 11, 16, 21, 26, and 31.
When the positive number n is divided by 7, 3 is left over, giving the equation n=7p+3, where pp is the quotient and the results are 3, 10, 17, 24, and 31.

Based on the previous two statements, it is possible to obtain a general formula for nn (of the form n=mx+r, where x is a divisor and rr is a remainder):
Since the two divisors above, 5 and 7, have a least common multiple of x, x=35.
In the previous two patterns, the remainder rr would be the first common integer, therefore r=31
N=35m+31 is the general formula based on both statements. Since k+n must be a multiple of 35, the smallest positive integer k that does so is 4; so,
n+4 = 35k+31+4 = 35(k+1)

Hence, B is the correct answer.

Approach Solution 2:

The leftover is 1 when n is divided by 5.
Therefore, n might have any of the following values: 1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, etc.

N divided by 7 leaves a residual of 3.
Therefore, n might be one of the following values: 3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, etc.
Therefore, it is clear that n might have a value of 31, 66, or any other number infinity.
Important: Since 35 is the Least Common Multiple of 7 and 5, we may deduce that each value of n will be 35 more than the previous value if we list all possible values for n.

Therefore, n might be equal to 31, 66, 101, 136, and so forth.

Check the available options.
Answer option A: The sum is NOT a multiple of 35 if we add 3 to any of these potential n-values.

GET RID OF A
Option B: If we add 4 to ANY of these potential n-values, the result will be a multiple of 35.
Therefore, 4 is the lowest number of k that will cause k+n to be a multiple of 35.

Hence, B is the correct answer.

Approach Solution 3:

Here, n is divided by 5 and 7, with 1 and 3 being the remainders. The number can be produced from the LCM of two (in this case, 2) numbers and the constant difference if the difference between the residual and the difference between the two is the same.
The required value is of the type A(LCM of 5 and 7) - constant difference = 35A - 4 since the constant difference, in this case, is 5-1 = 4 and 7-3 = 4.

So to obtain a multiple of 35, we would need to add 4 to 35A - 4.

Hence, B is the answer.

“When positive integer n is divided by 5, the remainder is 1. When n is" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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