By what least number should the given number be multiplied to get a perfect square number 2156

EXERCISE 3A

(1) Using the prime factorization method, find which of the following numbers are perfect squares:

(i) 441 = 3 × 3 × 7 × 7 = 32 × 72

(ii) 576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 = 26 × 32

(iii) 11025 = 5 × 5 × 3 × 3 × 7 × 7 = 52 × 32 × 72

(iv) 1176 = 2 × 2 × 2 × 3 × 7 × 7

(v) 5625 = 3 × 3 × 5 × 5 × 5 × 5 = 32 × 54

(vi) 9075 = 3 × 5 × 5 × 11 × 11

(vii) 4225 = 5 × 5 × 13 × 13 = 52 × 132

(viii) 1089 = 3 × 3 × 11 × 11 = 32 × 112

(2) Show that each of the following numbers is a perfect square. In each case, find the number whose square is the given number:

(i) 1225 = 5 × 5 × 7 × 7 = 52 × 72

Thus, 1225 is the product of pairs of equal factors.

∴ 1225 is a perfect square.

Also = (5 × 7)2 = (35)2

Hence, 35 is the number whose square is 1225.

(ii) 2601 = 3 × 3 × 17 × 17 = 32 × 172

Thus, 2601 is the product of pairs of equal factors.

∴ 2601 is a perfect square.

Also = (3 × 17)2 = (51)2

Hence, 51 is the number whose square is 2601.

(iii) 5929 = 7 × 7 × 11 × 11 = 72 × 112

Thus, 5929 is the product of pairs of equal factors.

∴ 5929 is a perfect square.

Also = (7 × 11)2 = (77)2

Hence, 77 is the number whose square is 5929.

(iv) 7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7 = 22 × 22 × 32 × 72

Thus, 7056 is the product of pairs of equal factors.

∴ 7056 is a perfect square.

Also = (2 × 2 × 3 × 7)2 = (84)2

Hence, 84 is the number whose square is 7056.

(v) 8281 = 7 × 7 × 13 × 13 = 72 × 132

Thus, 8281 is the product of pairs of equal factors.

∴ 8281 is a perfect square.

Also = (7 × 13)2 = (91)2

Hence, 91 is the number whose square is 8281.

(3) By what least number should the given number be multiplied to get a perfect square number? In each case, find the number whose square is the new number.

(i) 3675

Solution: Resolving 3675 into prime factors, we get

3675 = 3 × 5 × 5 × 7 × 7 = (3 × 52 × 72)

Thus, to get a perfect square number, the given number should be multiplied by 3.

New number = (32 × 52 × 72) = (3 × 5 × 7)2 = (105)2

Hence, the number whose square is the new number = 105.

(ii) 2156

Solution: Resolving 2156 into prime factors, we get

2156 = 2 × 2 × 7 × 7 × 11

Thus, to get a perfect square number, the given number should be multiplied by 11.

New number = (22 × 72 × 112) = (2 × 7 × 11)2 = (154)2

Hence, the number whose square is the new number = 154.

(iii) 3332

Solution: Resolving 3332 into prime factors, we get

3332 = 2 × 2 × 7 × 7 × 17

Thus, to get a perfect square number, the given number should be multiplied by 17.

New number = (22 × 72 × 172) = (2 × 7 × 17)2 = (238)2

Hence, the number whose square is the new number = 238.

(iv) 2925

Solution: Resolving 2925 into prime factors, we get

2925 = 3 × 3 × 5 × 5 × 13

Thus, to get a perfect square number, the given number should be multiplied by 13.

New number = (32 × 52 × 132) = (3 × 5 × 13)2 = (195)2

Hence, the number whose square is the new number = 195.

(v) 9075

Solution: Resolving 9075 into prime factors, we get

9075 = 3 × 5 × 5 × 11 × 11

Thus, to get a perfect square number, the given number should be multiplied by 3.

New number = (32 × 52 × 112) = (3 × 5 × 11)2 = (165)2

Hence, the number whose square is the new number = 165.

(vi) 7623

Solution: Resolving 7623 into prime factors, we get

7623 = 3 × 3 × 7 × 11 × 11

Thus, to get a perfect square number, the given number should be multiplied by 7.

New number = (32 × 72 × 112) = (3 × 7 × 11)2 = (231)2

Hence, the number whose square is the new number = 231.

(vii) 3380

Solution: Resolving 3380 into prime factors, we get

3380 = 2 × 2 × 5 × 13 × 13

Thus, to get a perfect square number, the given number should be multiplied by 5.

New number = (22 × 52 × 132) = (2 × 5 × 13)2 = (130)2

Hence, the number whose square is the new number = 130.

(viii) 2475

Solution: Resolving 2475 into prime factors, we get

2475 = 3 × 3 × 5 × 5 × 11

Thus, to get a perfect square number, the given number should be multiplied by 11.

New number = (32 × 52 × 112) = (3 × 5 × 11)2 = (165)2

Hence, the number whose square is the new number = 165.

(4) By what least number should the given number be divided to get a perfect square number? In each case, find the number whose square is the new number.

(i) 1575

Solution: Resolving 1575 into prime factors, we get

1575 = 3 × 3 × 5 × 5 × 7 = (32 × 52 × 7)

Thus, to get a perfect square number, the given number should be divided by 7.

New number obtained = (32 × 52) = (3 × 5)2 = (15)2

Hence, the number whose square is the new number = 15.

(ii) 9075

Solution: Resolving 9075 into prime factors, we get

9075 = 3 × 5 × 5 × 11 × 11 = (3 × 52 × 112)

Thus, to get a perfect square number, the given number should be divided by 3.

New number obtained = (52 × 112) = (5 × 11)2 = (55)2

Hence, the number whose square is the new number = 55.

(iii) 4851

Solution: Resolving 4851 into prime factors, we get

4851 = 3 × 3 × 7 × 7 × 11 = (32 × 72 × 11)

Thus, to get a perfect square number, the given number should be divided by 11.

New number obtained = (32 × 72) = (3 × 7)2 = (21)2

Hence, the number whose square is the new number = 21.

(iv) 3380

Solution: Resolving 3380 into prime factors, we get

3380 = 2 × 2 × 5 × 13 × 13 = (22 × 5 × 132)

Thus, to get a perfect square number, the given number should be divided by 5.

New number obtained = (22 × 132) = (2 × 13)2 = (26)2

Hence, the number whose square is the new number = 26.

(v) 4500

Solution: Resolving 4500 into prime factors, we get

4500 = 2 × 2 × 3 × 3 × 5 × 5 × 5 = (22 × 32 × 5 × 52)

Thus, to get a perfect square number, the given number should be divided by 5.

New number obtained = (22 × 32 × 52) = (2 × 3 × 5)2 = (30)2

Hence, the number whose square is the new number = 30.

(vi) 7776

Solution: Resolving 7776 into prime factors, we get

7776 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 = (22 × 22 × 2 × 3 × 32 × 32)

Thus, to get a perfect square number, the given number should be divided by 2 × 3.

New number obtained = (22 × 22 × 32 × 32) = (2 × 2 × 3 × 3)2 = (36)2

Hence, the number whose square is the new number = 36.

(vii) 8820

Solution: Resolving 8820 into prime factors, we get

8820 = 2 × 2 × 3 × 3 × 5 × 7 × 7 = (22 × 32 × 5 × 72)

Thus, to get a perfect square number, the given number should be divided by 5.

New number obtained = (22 × 32 × 72) = (2 × 3 × 7)2 = (42)2

Hence, the number whose square is the new number = 42.

(viii) 4056

Solution: Resolving 4056 into prime factors, we get

4500 = 2 × 2 × 2 × 3 × 13 × 13 = (22 × 2 × 3 × 132)

Thus, to get a perfect square number, the given number should be divided by 2 × 3.

New number obtained = (22 × 132) = (2 × 13)2 = (26)2

Hence, the number whose square is the new number = 26.

(5) Find the largest number of 2 digits which is a perfect square.

Ans: The largest 2 digits number is 99.

Square of 10 = 100 > 99, thus the number would be less than 10.

And the largest whole number less than 10 is 9.

Therefore, 9 × 9 = 81

(6) Find the largest number of 3 digits which is a perfect square.

Ans: The largest three digits number is 999. But 961 is a largest three digits number, is a perfect square.

961 = 31 × 31

Here, for easy to understand we take the before and after number of 31. Those are 30 and 32 respectively.

Now, 30 × 30 = 900 and 32 × 32 = 1024.

Hence, we can write 961 is largest three numbers has a perfect square.

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