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Example 6: Find the smallest square number that is exactly divisible by each one of the numbers,
6, 9 and 15.

Solution: Take LCM of 6, 9, 15 = 3 × 2 × 3 × 5 = 90.
3 6, 9, 15
But 90 is not a perfect square as 2 and 5 are not in pairs. 2, 3, 5

Thus, 90 must be multiplied by 2 × 5 = 10

Hence, the smallest square number exactly divisible by 6, 9 and 15 is 90 × 10 = 900.

Practice Time 5A

1. Identify whether the following numbers are perfect squares or not. Justify your answer.
(a) 243 (b) 729 (c) 2880 (d) 5625
2. Show that each of the following numbers is a perfect square. Also, find the number whose square
is the number given in question.

(a) 1444 (b) 1521 (c) 2025 (d) 3136
3. Find the smallest natural number by which 5808 should be multiplied to make it a perfect square.

4. Find the smallest natural number by which 9800 should be divided to make it a perfect square.
5. Find the smallest square number that is exactly divisible by each one of the numbers 2, 3, 6 and 10.
6. Find the smallest square number that is exactly divisible by each one of the numbers 8, 15 and 20.

Enrichment
Marie-Sophie Germain, a famous female mathematician, discovered that all numbers can be categorized as either
“happy” or “unhappy.” Let’s see how she came to her conclusion.
A happy number is one in which the sum of each digit squared eventually ends in the number 1.
For example, 13
2
2
2
2
1 + 3 = 1 + 9 = 10 and 1 + 0 = 1
Yes, 13 is a happy number.
Let’s check for 68
⇒ 6 + 8 = 36 + 64 = 100 ⇒ 1 + 0 + 0 ⇒ 1 + 0 + 0 = 1
2
2
2
2
2
Hence, 68 is also a happy number.
Sometimes many repetitions are required to reach 1.
For example, 44
2
2
2
2
4 + 4 = 32 ⇒ 3 + 2 = 9 + 4 = 13 Marie-Sophie Germain
⇒ 1 + 3 = 1 + 9 = 10 ⇒ 1 + 0 = 1 (1776-1831)
2
2
2
2
Hence, 44 is also a happy number
An unhappy number is one in which the sum of each digit squared creates an endless loop.
For example, 20
2
2 + 0 = 4 ⇒ 4 = 16 ⇒ 1 + 6 = 37
2
2
2
2
2
2
2
2
2
2
⇒ 3 + 7 = 58 ⇒ 5 + 8 = 89 ⇒ 8 + 9 = 145
2
⇒ 1 + 4 + 5 = 42 ⇒ 4 + 2 = 20
2
2
2
2
We returned to the starting number, 20, indicating an endless loop.
So, 20 is an unhappy number.
Check: l Is your roll number a happy number?
l Is your birthdate a happy number?
125 Squares and Square Roots