Page 41 - ICSE Math 8

Example 2: By what least number should 3,675 be multiplied to get a perfect square
number? Also, find the number whose square is the new number. 3 3675
Solution: Resolving 3,675 into its prime factors and forming pairs, we get 5 1225
2
2
3 × 5 × 5 × 7 × 7 = (3 × 5 × 7 ). 5 245
Thus, to get a perfect square number, the given number should be 7 49
7
7
multiplied by 3 as 3 does not exist in pair.
2
2
2
2
New number = (3 × 5 × 7 ) = (3 × 5 × 7) = (105) 2 1
Hence, the number whose square is the new number = 105.
Example 3: By what least number should 6300 be divided to get a perfect square
number? Find the number whose square is the new number. 3 6300
Solution: Resolving 6300 into its prime factors and forming pairs, we get 3 2100
2
2
2
3 × 3 × 7 × 5 × 5 × 2 × 2 = (3 × 7 × 5 × 2 ) 7 700
Thus, to get a perfect square number, the given number should be 5 100
divided by 7. 5 20
2
2
2
2
New number obtained = (3 × 5 × 2 ) = (3 × 5 × 2) = (30) 2 2 4
Hence, the number whose square is the new number = 30. 2 2
1
EXERCISE 3.1

1. Which of the following numbers are perfect squares? Give reasons.
(a) 6,241 (b) 625 (c) 921 (d) 249 (e) 1,024
2. Find the smallest number by which the given number must be multiplied to make it a perfect square.
(a) 882 (b) 432 (c) 1,331 (d) 845 (e) 3,698 (f) 700
3. Find the smallest number by which the given number must be divided to make it a perfect square.
(a) 8,112 (b) 3,920 (c) 3,971 (d) 1,568
4. Using prime factorization, find out which of the following numbers are perfect squares. Also find the
number whose square is given.
(a) 9,248 (b) 7,396 (c) 1,944 (d) 8,649 (e) 5,292 (f) 415
5. Find the greatest 3-digit number which is a perfect square.

6. Find the smallest 4-digit number which is a perfect square.

Properties of Perfect Squares
1. A number with 2, 3, 7 or 8 at its units place is never a perfect square.

For example, 42; 173; 3,287; 5,14,698, etc., are not perfect squares.

Conversely, property 1 implies that the numbers ending with digits other than 2, 3, 7 or 8 may or may not
be a perfect square. For example, 34, 155, 261, etc., are not perfect squares.
2. A number ending in an odd number of zeros Maths Info
is never a perfect square.
For example, 530 ; 46,000 ; 91,00,000 ; A perfect square when divided by 3 or 4 leaves the remainder
2
as 0 or 1. For example, (2) = 4, when divided by 3 (or 4) leaves
2,16,00,00,000, etc., are not perfect squares. the remainder 1 (or, 0); (3) = 9, when divided by 3 (or, 4) leaves
2
the remainder 0 (or, 1).

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