问题1.以下哪个数字是完美平方?
(i)484
解决方案：
Prime factorization of 484 is
484 = 2×2×11×11
By grouping prime factors
= (2×2) × (11×11)
Proper pairing of all the factors
Therefore, 484 is a perfect square.
(ii)625
解决方案：
Prime factorization of 625
625 = 5×5×5×5
By grouping the prime factors
= (5×5) × (5×5)
Proper pairing of all the factors
Therefore, 625 is a perfect square.
(iii)576
解决方案：
Prime factorization of 576
576 = 2×2×2×2×2×2×3×3
By grouping the prime factors
= (2×2) × (2×2) × (2×2) × (3×3)
Proper pairing of all the factors
Therefore, 576 is a perfect square.
(iv)941
解决方案：
Prime factorization of 941
941 = 941 × 1
941 itself is a prime factor.
Therefore, 941 is not a perfect square.
(v)961
解决方案：
Prime factorization of 961
961 = 31×31
By grouping the prime factors
= (31×31)
Proper pairing of all factors
Therefore, 961 is a perfect square.
(vi)2500
解决方案：
Prime factorization of
2500 = 2×2×5×5×5×5
By grouping the prime factors
= (2×2) × (5×5) × (5×5)
Proper pairing of all factors
Therefore, 2500 is a perfect square.
问题2。证明以下每个数字都是一个完美的平方。另外，在每种情况下，找到其平方是给定数字的数字：
(i)1156
解决方案：
Prime factorization of 1156
1156 = 2×2×17×17
By grouping prime factors
= (2×2) × (17×17)
Proper pairing of all factors
Therefore,1156 is a perfect square.
To find whose square the given number is
1156 = (2×17) × (17×2)
= 34 × 34
= (34)2
Therefore,1156 is a square of 34.
(ii)2025年
解决方案：
Prime factorization of 2025
2025 = 3×3×3×3×5×5
By grouping prime factors
= (3×3) × (3×3) × (5×5)
Proper pairing of all factors
Therefore, 2025 is a perfect square.
To find whose square the given number is
2025 = (3×3×5) × (3×3×5)
= 45 × 45
= (45)2
Therefore, 2025 is a square of 45.
(iii)14641
解决方案：
Prime factorization of 14641
14641 = 11×11×11×11
By grouping prime factors
= (11×11) × (11×11)
Proper pairing of all factors
Therefore,14641 is a perfect square.
To find whose square the given number is
14641 = (11×11) × (11×11)
= 121 × 121
= (121)2
Therefore,14641 is a square of 121.
(iv)4761
解决方案：
Prime factorization of 4761
4761 = 3×3×23×23
By grouping the prime factors
= (3×3) × (23×23)
Proper pairing of all factors
Therefore, 4761 is a perfect square.
To find whose square the given number is
4761 = (3×23) × (3×23)
= 69 × 69
= (69)2
Therefore, 4761 is a square of 69.
问题3.找出必须乘以给定数字的最小数，以使乘积是一个理想的平方：
(i)23805
解决方案：
Prime factorization of 23805
23805 = 3×3×23×23×5
By grouping the prime factors
= (3×3) × (23×23) × 5
Prime factor 5 is left out.
Multiplying by 5,
23805 × 5 = (3×3) × (23×23) × (5×5)
= (3×5×23) × (3×5×23)
= 345 × 345
= (345)2
Therefore, product is the square of 345.
(ii)12150
解决方案：
Prime factorization of 12150
12150 = 2×3×3×3×3×3×5×5
By grouping the prime factors
= 2×3 × (3×3) × (3×3) × (5×5)
Prime factor 2 and 3 are left out.
Multiplying by 2×3 = 6
12150 × 6 = 2×3 × (3×3) × (3×3) × (5×5) × 2 × 3
= (2×3×3×3×5) × (2×3×3×3×5)
= 270 × 270
= (270)2
Therefore, product is the square of 270.
(iii)7688
解决方案：
Prime factorization of 7688
7688 = 2×2×31×31×2
By grouping the prime factors
= (2×2) × (31×31) × 2
Prime factor 2 is left out.
Multiplying by 2
7688 × 2 = (2×2) × (31×31)× (2×2)
= (2×31×2) × (2×31×2)
= 124 × 124
= (124)2
Therefore, product is the square of 124.
问题4.找到给定数字必须除以的最小数字，以使所得的数字成为一个完美的平方：
(i)14283
解决方案：
Prime factorization of 14283
14283 = 3×3×3×23×23
By grouping the prime factors
= (3×3) × (23×23) × 3
Prime factor 3 is left out.
Dividing by 3
14283/3 = (3×3) × (23×23)
= (3×23) × (3×23)
= 69 × 69
= (69)2
Therefore, resultant is the square of 69.
(ii)1800
解决方案：
Prime factorization of 1800
1800 = 2×2×5×5×3×3×2
By grouping the prime factors
= (2×2) × (5×5) × (3×3) × 2
Prime factor 2 is left out.
Dividing by 2
1800/2 = (2×2) × (5×5) × (3×3)
= (2×5×3) × (2×5×3)
= 30 × 30
= (30)2
Therefore, resultant is the square of 30.
(iii)2904年
解决方案：
Prime factorization of 2904
2904 = 2×2×11×11×2×3
By grouping the prime factors
= (2×2) × (11×11) × 2 × 3
Prime factor 2 and 3 are left out.
Dividing by 6
2904/6 = (2×2) × (11×11)
= (2×11) × (2×11)
= 22 × 22
= (22)2
Therefore, resultant is the square of 22.
问题5.下列哪个数字是完美的平方?
11，12，16，32，36，50，64，79，81，111，121
解决方案：
11 = 1×11
11 is a prime number
Therefore, 11 is not a perfect square.
12 = 2×2×3
3 is left out in pairing of factors
Therefore, 12 is not a perfect square.
16 = 2×2×2×2
= (2×2) × (2×2)
Proper pairing of factors
Therefore,16 is a perfect square.
32 = 2×2×2×2×2
= (2×2) × (2×2) × 2
2 is left out in pairing of factors
Therefore, 32 is not a perfect square.
36 = 2×2×3×3
Proper pairing of factors
Therefore, 36 is a perfect square.
50 = 2×5×5
2 is left out in pairing of factors
Therefore, 50 is not a perfect square.
64 = 2×2×2×2×2×2
Proper pairing of factors
Therefore, 64 is a perfect square.
79 = 1×79
79 is a prime number
79 is not a perfect square.
81 = 3×3×3×3
Proper pairing of factors
Therefore, 81 is a perfect square.
111 = 1×111
111 it is a prime number.
Therefore, 111 is not a perfect square.
121 = 11×11
Proper pairing of factors
Therefore, 121 is a perfect square.
问题6.使用素数分解方法，找出以下哪个数字是理想平方?
189，225，2048，343，441，2961，11025，3549
解决方案：
189 prime factors are
189 = 3×3×3×7
As, proper pairing of factors is not there.
Therefore, it is not a perfect square
225 prime factors are
225 = (5×5) × (3×3)
Proper pairing of factors is there.
Therefore, it is a perfect square.
2048 prime factors are
2048 = (2×2) × (2×2) × (2×2) × (2×2) × (2×2) × 2
2 is left out from pairing of factors.
Therefore, 2048 is not a perfect square.
343 prime factors are
343 = (7×7) × 7
7 is left out from proper pairing of factors.
Therefore, 343 is not a perfect square.
441 prime factors are
441 = (7×7) × (3×3)
Proper pairing of factors is there.
Therefore, 441 is a perfect square.
2961 prime factors are
2961 = (3×3) × (3×3) × (3×3) × (2×2)
Proper pairing of factors is there.
Therefore, 2961 is a perfect square.
11025 prime factors are
11025 = (3×3) × (5×5) × (7×7)
Proper pairing of factors is there.
Therefore, 11025 is a perfect square.
3549 prime factors are
3549 = (13×13) × 3 × 7
Proper pairing of factors is not there.
Therefore, 3549 is not a perfect square.
问题7.在每种情况下，以下每个数字应乘以多少才能得到一个完美的平方?还找到其平方是新数字的数字。
(i)8820
解决方案：
Prime factorization of 8820
8820 = 2×2×3×3×7×7×5
By grouping the prime factors
= (2×2) × (3×3) × (7×7) × 5
Prime factor 5 is left out.
Multiplying by 5
8820 × 5 = (2×2) × (3×3) × (7×7) × (5×5)
= (2×3×7×5) × (2×3×7×5)
= 210 × 210
= (210)2
Therefore, new number is square of 210.
(ii)3675
解决方案：
Prime factorization of 3675
3675 = 5×5×7×7×3
By grouping the prime factors
= (5×5) × (7×7) × 3
Prime factor 3 is left out.
Multiplying by 3
3675 × 3 = (5×5) × (7×7) × (3×3)
= (5×7×3) × (5×7×3)
= 105 × 105
= (105)2
Therefore, new number is the square of 105.
(iii)605
解决方案：
Prime factorization of 605
605 = 5×11×11
By grouping the prime factors
= (11×11) × 5
Prime factor 5 is left out.
Multiply by 5
605 × 5 = (11×11) × (5×5)
= (11×5) × (11×5)
= 55 × 55
= (55)2
Therefore, new number is the square of 55.
(iv)2880
解决方案：
Prime factorization of 2880
2880 = 5×3×3×2×2×2×2×2×2
By grouping the prime factors
= (3×3) × (2×2) × (2×2) × (2×2) × 5
Prime factor 5 is left out.
Multiply by 5
2880 × 5 = (3×3) × (2×2) × (2×2) × (2×2) × (5×5)
= (3×2×2×2×5) × (3×2×2×2×5)
= 120 × 120
= (120)2
Therefore, new number is the square of 120.
(v)4056
解决方案：
Prime factorization of 4056
4056 = 2×2×13×13×2×3
By grouping the prime factors
= (2×2) × (13×13) × 2 × 3
Prime factors 2 and 3 are left out.
Multiplying by 6 we get,
4056 × 6 = (2×2) × (13×13) × (2×2) × (3×3)
= (2×13×2×3) × (2×13×2×3)
= 156 × 156
= (156)2
Therefore, new number is the square of 156.
(vi)3468
解决方案：
Prime factorization of 3468
3468 = 2×2×17×17×3
By grouping the prime factors
= (2×2) × (17×17) × 3
Prime factor 3 is left out.
Multiplying by 3 we get,
3468 × 3 = (2×2) × (17×17) × (3×3)
= (2×17×3) × (2×17×3)
= 102 × 102
= (102)2
Therefore, new number is the square of 102.
(vii)7776
解决方案：
Prime factorization of 7776
7776 = 2×2×2×2×3×3×3×3×2×3
By grouping the prime factors
= (2×2) × (2×2) × (3×3) × (3×3) × 2 × 3
Prime factors 2 and 3 are left out.
Multiplying by 6 we get,
7776 × 6 = (2×2) × (2×2) × (3×3) × (3×3) × (2×2) × (3×3)
= (2×2×3×3×2×3) × (2×2×3×3×2×3)
= 216 × 216
= (216)2
Therefore, the new number is square of 216.
问题8.在每种情况下，应将以下每个数字除以几以得到一个完美的平方?此外，找到其平方是新数字的数字。
(i)16562
解决方案：
Prime factorization of16562
16562 = 7×7×13×13×2
By grouping the prime factors
= (7×7) × (13×13) × 2
Prime factor 2 is left out.
Dividing by 2
16562/2 = (7×7) × (13×13)
= (7×13) × (7×13)
= 91 × 91
= (91)2
Therefore, the new number is square of 91.
(ii)3698
解决方案：
Prime factorization of 3698
3698 = 2×43×43
By grouping the prime factors
= (43×43) × 2
Prime factor 2 is left out.
Dividing by 2
3698/2 = (43×43)
= (43)2
Therefore, the new number is square of 43.
(iii)5103
解决方案：
Prime factorization of 5103
5103 = 3×3×3×3×3×3×7
By grouping the prime factors
= (3×3) × (3×3) × (3×3) × 7
Prime factor 7 is left out.
Dividing by 7
5103/7 = (3×3) × (3×3) × (3×3)
= (3×3×3) × (3×3×3)
= 27 × 27
= (27)2
Therefore, new number is the square of 27.
(iv)3174
解决方案：
Prime factorization of 3174
3174 = 2×3×23×23
By grouping the prime factors
= (23×23) × 2 × 3
Prime factor 2 and 3 are left out.
Dividing by 6
3174/6 = (23×23)
= (23)2
Therefore, new number is the square of 23.
(v)1575
解决方案：
Prime factorization for 1575
1575 = 3×3×5×5×7
By grouping the prime factors
= (3×3) × (5×5) × 7
Prime factor 7 is left out.
Dividing by 7
1575/7 = (3×3) × (5×5)
= (3×5) × (3×5)
= 15 × 15
= (15)2
Therefore, new number is square of 15.