BTGModeratorVI wrote: ↑

Sun Mar 15, 2020 12:42 pm

When integer n is divided by 15, the remainder is 5. Which of the following has a remainder of 10 when divided by 15 ?

I. 3n
II. 5n
III. 4n + 10

A. I Only
B. II Only
C. III Only
D. I & II Only
E. I, II & III

Answer: B
Source: Princeton Review

Say n = 15q + 5= 5(q + 3); where q is a positive integer

Let's take each statement one by one.

I. 3n: 3*5(3q + 1) = 15*(3q + 1). We see that 15*(3q + 1) is completely divisible by 15; thus the remainder = 0. Not the correct answer.

II. 5n: 5*5(3q + 1) = 75q + 25. We see that 75q is completely divisible by 15, and 25 divided by 15 will leave a remainder 10. Correct answer.

III. 4n + 10: 4*5(3q + 1) = 60q + 20. We see that 60q is completely divisible by 15, however, 20 divided by 15 will leave a remainder 5, not 10. Not the correct answer.

The correct answer: B

Hope this helps!

-Jay
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BTGModeratorVI wrote: ↑

Sun Mar 15, 2020 12:42 pm

When integer n is divided by 15, the remainder is 5. Which of the following has a remainder of 10 when divided by 15 ?

I. 3n
II. 5n
III. 4n + 10

A. I Only
B. II Only
C. III Only
D. I & II Only
E. I, II & III

Answer: B
Source: Princeton Review

Many integer properties questions can be solved by testing values

When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

When integer n is divided by 15, the remainder is 5.
So, some possible values of n are: 5, 20, 35, 50, 65, ....

Which of the following has a remainder of 10 when divided by 15 ?
Let's test a possible value of n.

Try n = 5
I. 3n = 3(5) = 15. When we divide 15 by 15, we get remainder 0. So, statement I does NOT satisfy the requirement.
ELIMINATE answer choices A, D and E

II. 5n = 5(5) = 25. When we divide 25 by 15, we get remainder 10. So, statement II DOES satisfy the requirement. This means we can't eliminate anything.

III. 4n + 10 = 4(5) + 10 = 30. When we divide 30 by 15, we get remainder 0. So, statement III does NOT satisfy the requirement.
ELIMINATE answer choice C

Answer: B

BTGModeratorVI wrote: ↑

Sun Mar 15, 2020 12:42 pm

When integer n is divided by 15, the remainder is 5. Which of the following has a remainder of 10 when divided by 15 ?

I. 3n
II. 5n
III. 4n + 10

A. I Only
B. II Only
C. III Only
D. I & II Only
E. I, II & III

Answer: B
Source: Princeton Review

We can create the equation in which Q = quotient:

n = 15Q + 5

Let’s now analyze each Roman Numeral:

I. 3n

3n = 45Q + 15

(45Q + 15)/15 = 3Q + 1

We see that 3n is evenly divisible by 15, so the remainder is 0.

II. 5n

5n = 45Q + 25

(45Q + 25)/15 = (45Q + 15)/15 + 10/15 + 3Q + 1 + 10/15

We see that the remainder is 10, so II is true.

III. 4n + 10

4n + 10 = 60Q + 20 + 10 = 60Q + 30

(60Q + 30)/15 = 4Q + 2

We see that 4n + 10 is evenly divisible by 15, so the remainder is zero.

Answer: B