When the Numbers 5, 7, 11 Divide a Positive Multiple of 17, the Remainders Left GMAT Problem Solving

Question: When the numbers 5, 7, 11 divide a positive multiple of 17, the remainders left are respectively 4, 6, and 10. Which positive multiple of 17 gives the least number that satisfies the given condition?

1. 315^th
2. 316th
3. 317th
4. 384th
5. 385th

Solution and Explanation

Approach Solution (1):

We can see that the difference between the divisor and remainder is constant = 1
5 – 4 = 1
7 – 6 = 1
11 – 10 = 1
So the least number will be 1 less than the LCM (5, 7, 11)
LCM (5, 7, 11) = 385, Least such number = 385 – 1 = 384
Since the number has to be a multiple of 17, it will be of the form 385n – 1 = 17k
The least such number is 5389 = 17 * 317

Correct option: C

Approach Solution (2):

In this question, we will use the Units Digit Analysis Method.
We will start with the first three answer choices.
1st positive multiple of 17 = 1 * 17 = 17
2nd positive multiple of 17 = 2 * 17 = 34
And so on
We are looking for the least positive multiple of 17
Given: 17m = 385k – 1
This can be written as: 385k = 17m + 1
On the LHS of the equation, the factor 385 multiplied by any non-negative integer k will yield a Units Digit of either 0 or 5.
Therefore, the RHS must yield a Units Digit of 0 and 5
This can also be read as: 17m + 1 must yield a Units Digit of 0 and 5
(1) 315th positive multiple of 17
(17) (315) + 1 (units digit of 6)

(2) 316th positive multiple of 17
(17) (316) + 1 (units digit of 3)

(3) 317th positive multiple of 17
(17) (317) + 1 (units digit of 0)

(4) 384th positive multiple of 17
(17) (384) + 1 (units digit of 9)

(5) 385th positive multiple of 17
(17) (385) + 1 (units digit of 6)

The only possible answer that could work is (c) 317th positive multiple of 17 = 5, 389

Correct option: C

Approach Solution (3):

Given:\(\frac{(17*Unkown)}{5}\)
Remainder: 4
Multiply each no's last digit by 7 if 17, we will get:

A. 315th: 7*5 = 35 =\(\frac{35}{5} \)(Perfectly divide, no remainder)
B. 316th: 7*6 = 42 =\(\frac{30}{5}\)(Perfectly divide, remainder= 2)
C. 317th: 7*7 = 49 =\(\frac{49}{5}\)(Perfectly divide, remainder= 4)
D. 384th: 7*4 = 28 =\(\frac{28}{5}\)(Perfectly divide, remainder= 3)
E. 385th: 7*5 = 35 =\(\frac{35}{5}\)(Perfectly divide, no remainder)

Correct option: C

“When the numbers 5, 7, 11 divide a positive multiple of 17, the remainders left are respectively 4, 6, and 10. Which positive multiple of 17 gives the least number that satisfies the given condition?”- 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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