Data Sufficiency

BTGmoderatorDC wrote: ↑

Sat Nov 26, 2022 6:38 pm

What is the remainder when n is divided by 26, given that n divided by 13 gives “a” as the quotient and “b” as the remainder? (a, b and n are positive integers)

(1) a is odd
(2) b = 3


OA C

Source: Veritas Prep

Target question: What is the remainder when n is divided by 26

Given: n divided by 13 gives “a” as the quotient and “b” as the remainder? (a, b and n are positive integers)
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2

So, we can taken the given information and write: n = 13a + b

Statement 1: a is odd
There's no information about b, so it will be impossible to determine the remainder when divided by 26.
Consider these two cases:
Case a: a = 1 and b = 2. In this case, n = (13)(1) + 2 = 15, so n divided by 26 leaves a remainder of 15
Case b: a = 1 and b = 3. In this case, n = (13)(1) + 3 = 16, so n divided by 26 leaves a remainder of 16
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b = 3
Consider these two contradictory cases:
Case a: a = 2 and b = 3. In this case, n = (13)(2) + 3 = 29, so n divided by 26 leaves a remainder of 3
Case b: a = 1 and b = 3. In this case, n = (13)(1) + 3 = 16, so n divided by 26 leaves a remainder of 16
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that a is odd, which means a = 2k + 1 for some integer k
Statement 2 tells us that b = 3

So, let's take the given information (n = 13a + b) and replace a with 2k + 1 and replace b with 3 to get:
n = 13(2k + 1) + 3
= 26k + 13 + 3
= 26k + 16
Here we can see that n is 16 greater than some multiple of 26, so when we divide n by 26, the remainder will be 16
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent