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Properties of Numbers

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Properties of Numbers

by minuit7 » Tue Aug 02, 2016 12:30 pm
I am seeking a clearer way to approach question #65 in The Official Guide for GMAT Quantitative Review 2017 than that which is provided in the answer section.

The question reads:
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?

The answer is provided but I think there must be a clearer and more intuitive approach to solving this problem.

Thanks!
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by Brent@GMATPrepNow » Tue Aug 02, 2016 12:35 pm
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
There's a nice rule that says, If, when N is divided by D, the remainder is R, then the possible values of N include: R, R+D, R+2D, R+3D,. . .

When n is divided by 5, the remainder is 1.
So, possible values of n are 1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, etc.

When n is divided by 7, the remainder is 3.
So, possible values of n are 3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, etc.

So, we can see that n could equal 31, or 66, or an infinite number of other values.

Important: Since the Least Common Multiple of 7 and 5 is 35, we can conclude that if we list the possible values of n, each value will be 35 greater than the last value.
So, n could equal 31, 66, 101, 136, and so on.

Check the answer choices....

Answer choice A: If we add 3 to any of these possible n-values, the sum is NOT a multiple of 35.
ELIMINATE A

Answer choice B: if we take ANY of these possible n-values, and add 4, the sum will be a multiple of 35.

So, the smallest value of k is 4 such that k+n is a multiple of 35.

Answer = B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Jeff@TargetTestPrep » Wed Aug 03, 2016 8:41 am
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
We can find the value of n first by just strategically find values that when divided by 5 have a reminder of 1. For example, since the remainder is 1 when n is divided by 5, n will be a [(multiple of 5) + 1] and thus must be one of the following numbers:

1, 6, 11, 16, 21, 26, 31, ...

Now we have to find out which of these numbers when divided by 7, have a remainder of 3.

1/7 = 0 remainder 1

6/7 = 0 remainder 6

11/7 = 0 remainder 6

6/7 = 1 remainder 4

16/7 = 2 remainder 2

21/7 = 3 remainder 0

26/7 = 3 remainder 5

31/7 = 4 remainder 3

We can see that 31 is the smallest value of n that satisfies the requirement. So we must determine the value of k such that k + n is a multiple of 35. Obviously, since 4 + 31 = 35 and 35 is a multiple of 35, then the smallest positive integer value of k is 4.

Answer:B

Jeffrey Miller
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by Matt@VeritasPrep » Thu Aug 04, 2016 8:29 pm
I'd think of it this way:

Let's say that n = 5*(some integer) + 1 and n = 7*(some other integer) + 3.

Naming "some integer" x and "some other integer" y, we have

5x + 1 = n
7y + 3 = n

5x + 1 = 7y + 3
5x = 7y + 2

Now let's look for the smallest positive solution to this equation: something where a multiple of 5 is also 2 greater than a multiple of 7. Checking our multiples of 5 and our "2 greater than a multiple of 7s", we have

5, 10, 15, 20, 25, 30 ...
2, 9, 16, 23, 30, ...

So the first number on both lists is 30, making that the smallest solution.

That gives us 5x = 30, and since n = 5x + 1, we know n = 31.

From here, k + n = 35, and with n = 31, we must have k = 4.
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