Let n be the smallest positive integer that is a multiple of 15 and leaves a remainder equ… — GMAT

Let $n$ be the smallest positive integer that is a multiple of 15 and leaves a remainder equivalent to $-2$ when divided by 7. What is the value of $n$ ?

A
30
75Answer
C
45
D
105
E
60

Answer

The correct value is 75. Since

n

is a multiple of 15,

n = 15k

. A remainder of

-2

when divided by 7 means

n \equiv 5 \pmod 7

. Substituting

n = 15k

gives

15k \equiv 5 \pmod 7

, which simplifies to

k \equiv 5 \pmod 7

since

15 \equiv 1 \pmod 7

. The smallest positive integer value for

k

is 5, giving

n = 15 \times 5 = 75

Step-by-Step Solution

Express

n

as a multiple of 15

n = 15k

for some positive integer

k

Since

n

is a multiple of 15, it must be of the form

15k

Convert the negative remainder condition into a standard positive remainder modulo 7

n \equiv -2 \equiv 5 \pmod 7

A remainder of

-2

modulo 7 is equivalent to

7 + (-2) = 5

Solve the modular congruence for

k

k = 5

Substituting

n = 15k

into

15k \equiv 5 \pmod 7

gives

1k \equiv 5 \pmod 7

because

15 \equiv 1 \pmod 7

. The smallest positive integer solution is

k = 5

Calculate the value of

n

n = 15 \times 5 = 75

Multiplying 15 by

k = 5

yields 75.

Key Concept

Modular Arithmetic and Remainder Properties

Estimated Time:1m 30s

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