CAT 2026 Slot 1 Quant: Numbers Questions with Solutions

3 questions: Q2, Q13, Q22. Solve each one here: tap the answer, press Check, then open the solution.

Question 2Number System

TITA (type the answer) · +3 / 0

For positive integers $a$, $b$ and $c$, if $ab + bc = 13$ and $bc + ca = 45$, what is the minimum possible value of $abc$?

Your answer

Answer and solution

Correct answer: 36

Solution

$b(a + c) = 13$. Since 13 is prime and $a + c \ge 2$: $b = 1,\; a + c = 13$.
$c(a + b) = 45 \Rightarrow c(a + 1) = 45 \Rightarrow c(14 - c) = 45$
$c^2 - 14c + 45 = 0 \Rightarrow c = 5$ or $9$
$c = 5, a = 8$: $abc = 40$; $\;c = 9, a = 4$: $abc = 36$
Minimum $abc = 36$

Question 13Number System – Divisors

MCQ · +3 / −1

The number of divisors of $2^4 \times 3^5 \times 5^3 \times 7^2$ that are of the form $4r + 1$, where $r$ is a non-negative integer, is:

Answer and solution

Correct answer: A

Solution

A divisor of the form $4r + 1$ is odd, so it has no factor 2: $3^a \cdot 5^b \cdot 7^c$.
$5^b \equiv 1 \pmod 4$ always → 4 choices of $b$.
$3^a \equiv 3$ or $1$ and $7^c \equiv 3$ or $1$ (odd/even powers). Need $a + c$ even.
$a \in \{0..5\}$: 3 even, 3 odd; $c \in \{0,1,2\}$: 2 even, 1 odd → $3 \times 2 + 3 \times 1 = 9$
Total $= 9 \times 4 = 36$

Question 22Number System – Digits

MCQ · +3 / −1

For a 4-digit number greater than 1000, the sum of the thousands, hundreds and tens digits is 13, and the sum of the hundreds, tens and units digits is 15. The tens digit is 4 more than the units digit. What is the difference between the largest and smallest possible values of the number?

Answer and solution

Correct answer: A

Solution

Let the digits be $a, b, c, d$. $a + b + c = 13$, $b + c + d = 15$, $c = d + 4$.
Subtracting: $d - a = 2 \Rightarrow d = a + 2,\; c = a + 6$
$a + b + a + 6 = 13 \Rightarrow b = 7 - 2a$
$a = 1$: 1573 · $a = 2$: 2384 · $a = 3$: 3195
Difference $= 3195 - 1573 = 1622$

An expected paper by Cetking, not the actual CAT paper. CAT is conducted by the IIMs.