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CAT Number System questions with solutions

Factors, divisors, factorials and number theory. 19 questions from our 2016–2025 CAT pattern papers. Each has the answer and a full solution.

CAT 2025

Find the remainder when 7⁴⁵ is divided by 11.CAT 2025 Slot 1 · hard
Find the remainder when 23⁵¹ is divided by 100.CAT 2025 Slot 3 · hard

CAT 2024

A number, when divided by 7, leaves a remainder of 5, and when divided by 11, leaves a remainder of 3. Find the remainder when the number is divided…CAT 2024 Slot 1 · hard

CAT 2023

What is the remainder when 29¹⁵⁶ is divided by 17?CAT 2023 Slot 3 · hard

CAT 2022

What is the remainder when 7¹⁰⁰ is divided by 25?CAT 2022 Slot 2 · hard
Find the number of positive integers less than 200 that are divisible by 3 or by 5, but not by 15.CAT 2022 Slot 3 · hard

CAT 2021

How many ordered pairs of positive integers (x, y) satisfy x² − y² = 175?CAT 2021 Slot 1 · hard
How many ordered pairs of positive integers (x, y) satisfy xy + x + y = 104?CAT 2021 Slot 3 · hard

CAT 2020

How many divisors of N = 2⁴ × 3³ × 5² are perfect squares?CAT 2020 Slot 1 · medium
How many ordered pairs of positive integers (x, y) satisfy 1/x + 1/y = 1/12?CAT 2020 Slot 2 · hard
How many ordered pairs of positive integers (a, b) satisfy a² − b² = 45?CAT 2020 Slot 3 · hard

CAT 2019

How many even positive divisors does 720 have?CAT 2019 Slot 1 · medium

CAT 2018

How many ordered pairs of integers (x, y) satisfy x² − y² = 96?CAT 2018 Slot 1 · hard
How many positive divisors of 1080 are multiples of 6?CAT 2018 Slot 1 · medium
How many ordered pairs of positive integers (x, y) satisfy 1/x + 1/y = 1/6?CAT 2018 Slot 2 · hard
How many zeros are there at the end of 125! (125 factorial)?CAT 2018 Slot 2 · medium

CAT 2017

How many ordered pairs (x, y) of positive integers satisfy 1/x + 1/y = 1/12?CAT 2017 Slot 1 · hard
What is the largest integer n such that 12ⁿ divides 50! (50 factorial)?CAT 2017 Slot 2 · hard

CAT 2016

How many divisors of 1800 are perfect squares?CAT 2016 Slot 1 · medium