By what number should each of the following numbers be multiplied to get a perfect square in each case?
By what number should each of the following numbers be multiplied to get a perfect square in each case? Also, find the number whose square is the new number.
(i) 8820
(ii) 3675
(iii) 605
(iv) 2880
(v) 4056
(vi) 3468
(vii) 7776
Factorising each number.
(i) 8820 = 2 x 2 x 3 x 3 x 5 x 7 x 7
Grouping them into pairs of equal factors:
8820 = (2 x 2) x (3 x 3) x (7 x 7) x 5
The factor, 5 is not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 8820 must be multiplied by 5 for it to be a perfect square.
The new number would be (2 x 2) x (3 x 3) x (7 x 7) x (5 x 5).
Furthermore, we have:
(2 x 2) x (3 x 3) x (7 x 7) x (5 x 5) = (2 x 3 x 5 x 7) x (2 x 3 x 5 x 7)
Hence, the number whose square is the new number is:
2 x 3 x 5 x 7 = 210
(ii) 3675 = 3 x 5 x 5 x 7 x 7
Grouping them into pairs of equal factors:
3675 = (5 x 5) x (7 x 7) x 3
The factor, 3 is not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 3675 must be multiplied by 3 for it to be a perfect square.
The new number would be (5 x 5) x (7 x 7) x (3 x 3).
Furthermore, we have:
(5 x 5) x (7 x 7) x (3 x 3) = (3 x 5 x 7) x (3 x 5 x 7)
Hence, the number whose square is the new number is:
3 x 5 x 7 = 105
(iii) 605 = 5 x 11 x 11
Grouping them into pairs of equal factors:
605 = 5 x (11 x 11)
The factor, 5 is not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 605 must be multiplied by 5 for it to be a perfect square.
The new number would be (5 x 5) x (11 x 11).
Furthermore, we have:
(5 x 5) x (11 x 11) = (5 x 11) x (5 x 11)
Hence, the number whose square is the new number is:
5 x 11 = 55
(iv) 2880 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5
Grouping them into pairs of equal factors:
2880 = (2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x 5
There is a 5 as the leftover. For a number to be a perfect square, each prime factor has to be paired. Hence, 2880 must be multiplied by 5 to be a perfect square.
The new number would be (2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x (5 x 5).
Furthermore, we have:
(2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x (5 x 5) = (2 x 2 x 2 x 3 x 5) x (2 x 2 x 2 x 3 x 5)
Hence, the number whose square is the new number is:
2 x 2 x 2 x 3 x 5 = 120
(v) 4056 = 2 x 2 x 2 x 3 x 13 x 13
Grouping them into pairs of equal factors:
4056 = (2 x 2) x (13 x 13) x 2 x 3
The factors at the end, 2 and 3 are not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 4056 must be multiplied by 6 (2 x 3) for it to be a perfect square.
The new number would be (2 x 2) x (2 x 2) x (3 x 3) x (13 x 13).
Furthermore, we have:
(2 x 2) x (2 x 2) x (3 x 3) x (13 x 13) = (2 x 2 x 3 x 13) x (2 x 2 x 3 x 13)
Hence, the number whose square is the new number is:
2 x 2 x 3 x 13 = 156
(vi) 3468 = 2 x 2 x 3 x 17 x 17
Grouping them into pairs of equal factors:
3468 = (2 x 2) x (17 x 17) x 3
The factor 3 is not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 3468 must be multiplied by 3 for it to be a perfect square.
The new number would be (2 x 2) x (17 x 17) x (3 x 3).
Furthermore, we have:
(2 x 2) x (17 x 17) x (3 x 3) = (2 x 3 x 17) x (2 x 3 x 17)
Hence, the number whose square is the new number is:
2 x 3 x 17 = 102
(vii) 7776 = 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3
Grouping them into pairs of equal factors:
7776 = (2 x 2) x (2 x 2) x (3 x 3) x (3 x 3) x 2 x 3
The factors, 2 and 3 at the end are not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 7776 must be multiplied by 6 (2 x 3) for it to be a perfect square.
The new number would be (2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x (3 x 3) x (3 x 3).
Furthermore, we have:
(2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x (3 x 3) x (3 x 3) = (2 x 2 x 2 x 3 x 3 x 3) x (2 x 2 x 2 x 3 x 3 x 3)
Hence, the number whose square is the new number is:
2 x 2 x 2 x 3 x 3 x 3 = 216
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