By What numbers should each of the following be divided to get a perfect square in each case? Also, find the number whose square is the new number.
(i) 16562
(ii) 3698
(iii) 5103
(iv) 3174
(v) 1575
(i) 16562
16562 = (7 × 7) × (13 × 13) × 2
= (7 × 7) × (13 × 13)
= (7 × 13) × (7 × 13)
= 91 × 91
= 912
Therefore, the resultant is the square of 91
3698 = 2 × (43 × 43)
= 43 × 43
= 432
Therefore, the numbers must be divided by 2 and resultant is square of 43
5103 = (3 × 3) × (3 × 3) × (3 × 3) × 7
= (3 × 3 × 3) × (3 × 3 × 3)
= 27 × 27
= 272
Therefore, the number must be divided by 7 and resultant is square of 27
3174 = 2 × 3 × (23 × 23)
= 23 × 23
= 232
Therefore, the number must be divided by 6 and the resultant is square of 23
1575 = 3 × 3 × 5 × 5 × 7
= 3 × 3 × 5 × 5
= (3 × 5) × (3 × 5)
= 15 × 15
= 152
Therefore, the number must be divided by 7 and the resultant is square of 15