Page 47 - ICSE Math 8

Example 14: Find the smallest square number divisible by each one of the 2 6, 10, 12, 15
numbers 6, 10, 12, 15. 2 3, 5, 6, 15
Solution: LCM of 6, 10, 12, 15 = 2 × 2 × 3 × 5 = 60 3 3, 5, 3, 15
Now write 60 as a product of prime factors of the given 5 1, 5, 1, 5
numbers. 1, 1, 1, 1
So, 60 = 2 × 2 × 3 × 5

Factors 3 and 5 are left unpaired. 2 60
To make it a perfect square, we need to multiply 60 by 3 × 5. 2 30
So, 60 × (3 × 5) = 60 × 15 = 900 3 15
Hence, the smallest square number divisible by 6, 10, 12, 15 is 900. 5 5
1

Example 15: The product of two numbers is 256. If one number is 4 times the other, find the numbers.
Solution: Product of two numbers = 256
Let one number be x. 2 1,024
256 2 512
Then the other number = 2 256
x
256 2 128
According to the question, x = 4 × 2 64
2
fi x = 4 × 256 x 2 32
fi x = 4 256× = 1,024 2 16

= 22 22 22 22 22× × × × × × × × × = 32 2 8
2 4
= 2 × 2 × 2 × 2 × 2 2 2
256
\ One number = 32 and other number = = 8 1
32

EXERCISE 3.3

1. What are the possible units digits of the square roots of the following numbers? Which of these have even
square roots?
(a) 9,604 (b) 99,225 (c) 1,06,276 (d) 1,76,400

2. Find the square root of the following using prime factorization.
(a) 576 (b) 1,444 (c) 2,401 (d) 3,364 (e) 4,761
3. In each of the following, find the smallest number by which it should be multiplied to get a perfect square.
Find the square root of the perfect squares so obtained.
(a) 392 (b) 216 (c) 3,312 (d) 3,698
4. In each of the following, find the smallest number by which it should be divided to get a perfect square.
Find the square root of the perfect squares so obtained.
(a) 4,050 (b) 4,332 (c) 53,016 (d) 38,440 (e) 5,832

5. The product of two numbers is 2,916. If one number is 36 times the other, find the numbers.
6. A group of boys collected 256 marbles for some project. The contribution of each boy was equal to the
number of boys in the group. How many boys were there in the group?
2
7. The area of a square garden is 4,624 m . What is its dimension?

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