CAT Problems on perfect squares and perfect cubes - Practice Questions & MCQ

Introduction

The concept of perfect squares and perfect cubes is an important topic in the number system. Perfect squares are numbers that can be expressed as the product of two equal integers, while perfect cubes are numbers that can be expressed as the product of three equal integers.

Classification of Numbers

• Perfect Squares: Numbers such as 4, 9, 16, 25, etc., which have exact square roots, are considered perfect squares.
• Perfect Cubes: Numbers such as 8, 27, 64, 125, etc., which have exact cube roots, are considered perfect cubes.

Properties of Perfect Squares

• Perfect squares are always non-negative.
• The unit digit of a perfect square can only be 0, 1, 4, 5, 6

Properties of Perfect Cubes

• Perfect cubes can be both positive and negative.
• The unit digit of a perfect cube can only be 0, 1, or 8.

Solved Examples:

Question 1: Find the least perfect square number which is divisible by 4, 9, and 10.

Solution: To find the least perfect square number that is divisible by 4, 9, and 10, we must first find the least common multiple (LCM) of these numbers, as this will be the smallest number that all three can divide evenly into.

The prime factors of 4, 9, and 10 are:

4 = 2 * 2

9 = 3 * 3

10 = 2 * 5

Combining these gives us the least common multiple (LCM) as the product of the highest power of all primes occurring in the factorization of these numbers.

So the LCM is 2^2 * 3^2 * 5 = 4 * 9 * 5 = 180.

However, 180 is not a perfect square. For a number to be a perfect square, each prime factor must occur in pairs. Here, the prime factor 5 is not in pair. Therefore, to make this a perfect square, we need to multiply it by another 5.

So, the least perfect square number which is divisible by 4, 9, and 10 is 180 * 5 = 900.

Question 2: Find the largest perfect cube number which is a factor of 189.

First, we find the prime factorization of 189:

189 = 3 * 3 * 3 * 7.

From this factorization, we can see that the largest perfect cube that divides 189 is 3^3 = 27. The number 7 can't form a perfect cube because it doesn't have two other 7s to pair with.

Tips and Tricks

• To find if a number is a perfect square, check if its prime factorization has every prime factor occurring an even number of times.
• For finding the last digit of a perfect square, you can directly find the last digit of the given number.
• For solving questions on perfect squares, it is helpful to have the squares of numbers up to 30 (e.g., 1^2 = 1, 2^2 = 4, 3^2 = 9, etc.) memorised.
• For solving questions on perfect cubes, it is helpful to have the cubes of numbers up to 10 (e.g., 1^3 = 1, 2^3 = 8, 3^3 = 27, etc.) memorised.