For integers m and k, is the remainder when N = k(m+7) (m+8) divided b

Question

For integers m and k, is the remainder when N = k(m+7)(m+8) divided by 3 equal to 0?

(1) k = m - 3
(2) m is divisible by 3.

gmatophobia · Answer

The question provides us with a clue that is worth taking note of -

(m+7) (m+8) are consecutive integers.
 We know that the product of three consecutive numbers is divisible by 3 as one of three numbers is divisible by 3.

Statement 1

(1) k=m-3

m - 3 will have the same remainder as (m + 6), this is because the remainder repeats in an interval of three (as the divisor = 3).

For example : 
Assume that : Remainder( \frac{m - 3 }{ 3} ) = 1

Remainder( \frac{m - 3 + 1 }{ 3} )= Remainder( \frac{m - 3 }{ 3} ) + Remainder( \frac{1 }{ 3} ) = Remainder( \frac{1 + 1}{3} ) = 2

Remainder( \frac{m - 3 + 2 }{ 3} )= Remainder( \frac{m - 3 }{ 3} ) + Remainder( \frac{2 }{ 3} ) = Remainder( \frac{1 + 2}{3} ) = 0

Remainder( \frac{m - 3 + 3 }{ 3} )= Remainder( \frac{m - 3 }{ 3} ) + Remainder( \frac{3 }{ 3} ) = Remainder( \frac{1 + 0}{3} ) = 1

Therefore, Remainder( \frac{m - 3 }{ 3} ) = Remainder( \frac{m - 3 + 3 }{ 3} ) = Remainder( \frac{m }{ 3} )

Similarly adding a further 3 wouldn't change the remainder

Remainder( \frac{m }{ 3} ) = Remainder( \frac{m+3}{ 3} ) = Remainder( \frac{m+6}{ 3} )

Hence we can conclude that

Remainder( \frac{k}{3} ) = Remainder( \frac{m+6}{ 3} )

Thus, among k, (m+7), and (m+8) one of the numbers will be a multiple of 3. This statement is sufficient to conclude that N is a multiple of 3.

We can eliminate options B, C, and E.

Statement 2

(2) m is divisible by 3.

The information provided helps us conclude the following -

Remainder( \frac{(m+7)}{3} ) = 1 
 Remainder( \frac{(m+8)}{3} ) = 2

If k is a multiple of 3 → N is a multiple of 3. The response to the question "is the remainder when N = k(m+7) (m+8) divided by 3 equal to 0" is Yes.

If k is not a multiple of 3 → N is a not multiple of 3. The response to the question "is the remainder when N = k(m+7) (m+8) divided by 3 equal to 0" is No.

Hence, this statement alone is not sufficient.

We can eliminate D.

Option A

aditya khanna · Answer

K=M-3
K=(M-3+9-9)
K=(M+6)-9
N = ((M+6)-9)(m+7)(m+8))
N= (m+7)(m+8)(m+6) -9(m+7)(m+8) 
Three consecutive numbers are divisible by 3! , hence N is divisible by 3

For integers m and k, is the remainder when N = k(m+7) (m+8) divided b

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GMAT Data Sufficiency (DS) Questions

For integers m and k, is the remainder when N = k(m+7) (m+8) divided b

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