Q.1. Using the prime factorisation method, find which of the following numbers are perfect squares:
(i) 441
(ii) 576
(iii) 11025
(iv) 1176
(v) 5625
(vi) 9075
(vii) 4225
(viii) 1089
Ans. A perfect square can always be expressed as a product of equal factors.
(i) Resolving into prime factors: 441=49×9=7×7×3×3=7×3×7×3=21×21=(21)2
Thus, 441 is a perfect square.
(ii) Resolving into prime factors:
576=64×9=8×8×3×3=2×2×2×2×2×2×3×3
=24×24=(24)2
Thus, 576 is a perfect square.
(iii) Resolving into prime factors:
11025=441×25=49×9×5×5
=7×7×3×3×5×5
=7×5×3×7×5×3
=105×105=(105)2
Thus, 11025 is a perfect square.
(iv) Resolving into prime factors:
1176=7×168=7×21×8=7×7×3×2×2×2
1176 cannot be expressed as a product of two equal numbers. Thus, 1176 is not a perfect square.
(v) Resolving into prime factors:
5625=225×25=9×25×25=3×3×5×5×5×5=3×5×5×3×5×5=75×75=(75)2
Thus, 5625 is a perfect square.
(vi) Resolving into prime factors:
9075=25×363=5×5×3×11×11=55×55×3
9075 is not a product of two equal numbers. Thus, 9075 is not a perfect square.
(vii) Resolving into prime factors:
4225=25×169=5×5×13×13=5×13×5×13=65×65=(65)2
Thus, 4225 is a perfect square.
(viii) Resolving into prime factors:
1089=9×121=3×3×11×11=3×11×3×11=33×33=(33)2
Thus, 1089 is a perfect square.
Q.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
(ii) 2601
(iii) 5929
(iv) 7056
(v) 8281
Ans. A perfect square is a product of two perfectly equal numbers.
(i) Resolving into prime factors:
1225=25×49=5×5×7×7=5×7×5×7=35×35=(35)2
Thus, 1225 is the perfect square of 35.
(ii) Resolving into prime factors:
2601=9×289=3×3×17×17=3×17×3×17=51×51=(51)2
Thus, 2601 is the perfect square of 51.
(iii) Resolving into prime factors:
5929=11×539=11×7×77=11×7×11×7=77×77=(77)2
Thus, 5929 is the perfect square of 77.
(iv) Resolving into prime factors:
7056=12×588=12×7×84=12×7×12×7=(12×7)2=(84)2
Thus, 7056 is the perfect square of 84.
(v) Resolving into prime factors:
8281=49×169=7×7×13×13=7×13×7×13=(7×13)2=(91)2
Thus, 8281 is the perfect square of 91.
Q.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) 3975
(ii) 2156
(iii) 3332
(iv) 2925
(v) 9075
(vi) 7623
(vii) 3380
(viii) 2475
Ans. 1. Resolving 3675 into prime factors:
3675=3×5×5×7×73675=3×5×5×7×7
Thus, to get a perfect square, the given number should be multiplied by 3.
New number= (32×52×72)=(3×5×7)2=(105)2
Hence, the new number is the square of 105.
2. Resolving 2156 into prime factors:
2156=2×2×7×7×11=(22×72×11)
Thus to get a perfect square, the given number should be multiplied by 11.
New number =(22×72×112)=(2×7×11)2=(154)2
Hence, the new number is the square of 154.
3. Resolving 3332 into prime factors:
3332=2×2×7×7×17=22×72×17
Thus, to get a perfect square, the given number should be multiplied by 17.
New number =(22×72×172)=(2×7×17)2=(238)2
Hence, the new number is the square of 238.
4. Resolving 2925 into prime factors:
2925=3×3×5×5×13=32×52×13
Thus, to get a perfect square, 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 is 195.
5. Resolving 9075 into prime factors:
9075=3×5×5×11×11=3×52×112
Thus, to get a perfect square, the given number should be multiplied by 3.
New number =(32×52×112)=(3×5×11)2=(165)2
Hence, the new number is the square of 165.
6. Resolving 7623 into prime factors:
7623=3×3×7×11×11=32×7×112
Thus, to get a perfect square, the given number should be multiplied by 7.
New number =(32×72×112)=(3×7×11)2=(231)2=(32×72×112)=(3×7×11)2=(231)2
Hence, the number whose square is the new number is 231.
7. Resolving 3380 into prime factors:
3380=2×2×5×13×13=22×5×132
Thus, to get a perfect square, the given number should be multiplied by 5.
New number =(22×52×132)=(2×5×13)2=(130)2
Hence, the new number is the square of 130.
8. Resolving 2475 into prime factors:
2475=3×3×5×5×11=32×52×11
Thus, to get a perfect square, the given number should be multiplied by 11.
New number =(32×52×112)=(3×5×11)2=(165)2
Hence, the new number is the square of 165.
Q.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
(ii) 9075
(iii) 4851
(iv) 3380
(v) 4500
(vi) 7776
(vii) 8820
(viii) 4056
Ans. (i) Resolving 1575 into prime factors:
1575=3×3×5×5×7=32×52×7
Thus, to get a perfect square, the given number should be divided by 7
New number obtained=(32×52)=(3×5)2=(15)2
Hence, the new number is the square of 15
(ii) Resolving 9075 into prime factors:
9075=3×5×5×11×11=3×52×112
Thus, to get a perfect square, the given number should be divided by 3
New number obtained=(52×112)=(5×11)2=(55)2
Hence, the new number is the square of 55
(iii) Resolving 4851 into prime factors:
4851=3×3×7×7×11=32×72×11
Thus, to get a perfect square, the given number should be divided by 11
New number obtained=(32×72)=(3×7)2=(21)2
Hence, the new number is the square of 21
(iv) Resolving 3380 into prime factors:
3380=2×2×5×13×13=22×5×132
Thus, to get a perfect square, the given number should be divided by 5
New number obtained=(22×132)=(2×13)2=(26)2
Hence, the new number is the square of 26
(v) Resolving 4500 into prime factors:
4500=2×2×3×3×5×5×5=22×32×52×5
Thus, to get a perfect square, the given number should be divided by 5
New number obtained=(22×32×52)=(2×3×5)2=(30)2
Hence, the new number is the square of 30
(vi) Resolving 7776 into prime factors:
7776=2×2×2×2×2×3×3×3×3×3=22×22×2×32×32×3
Thus, to get a perfect square, the given number should be divided by 6 whish is a product of 2 and 3
New number obtained=(22×22×32×32)=(2×2×3×3)2=(36)2
Hence, the new number is the square of 36
(vii) Resolving 8820 into prime factors:
8820=2×2×3×3×5×7×7=22×32×5×72
Thus, to get a perfect square, the given number should be divided by 5
New number obtained=(22×32×72)=(2×3×7)2=(42)2
Hence, the new number is the square of 42
(viii) Resolving 4056 into prime factors:
4056=2×2×2×3×13×13=22×2×3×132
Thus, to get a perfect square, the given number should be divided by 6, which is a product of 2 and 3
New number obtained=(22×132)=(2×13)2=(26)2
Hence, the new number is the square of 26
Q.5. Find the largest number of 2 digits which is a perfect square.
Ans. The first three digit number (100) is a perfect square. Its square root is 10.
The number before 10 is 9.
Square of 9 = (9)2=81= (9)2=81
Thus, the largest 2 digit number that is a perfect square is 81.
Q.6. Find the largest number of 3 digits which is a perfect square.
Ans. The largest 3 digit number is 999.
The number whose square is 999 is 31.61.
Thus, the square of any number greater than 31.61 will be a 4 digit number.
Therefore, the square of 31 will be the greatest 3 digit perfect square.
312=31×31=961
