CAT 2025 Slot 2 QA Q11 Number Systems

Question 11 of the Quantitative Ability (QA) section, CAT 2025 Slot 2, with CAT's official answer and a worked solution.

The number of divisors of \( \left( 2 ^ { 6 } \times 3 ^ { 5 } \times 5 ^ { 3 } \times 7 ^ { 2 } \right) \), which are of the form \( ( 3 r + 1 ) \), where \( r \) is a non-negative integer, is

A
\( 42 \)
B
\( 36 \)
C
\( 56 \)
D
\( 24 \)

Answer: (A) 42 Number Systems MCQ

Solution

A divisor \(\equiv 1 \pmod 3\) cannot contain 3. Modulo 3: \(2 \equiv -1\), \(5 \equiv -1\), \(7 \equiv 1\).
So for \(2^a 5^c 7^e\) we need \(a + c\) even; \(e \in \{0,1,2\}\) is free (3 ways).
\(a \in 0..6\), \(c \in 0..3\): both even \(4 \times 2 = 8\); both odd \(3 \times 2 = 6\). Total 14 pairs.
Count \(= 14 \times 3 = 42\).