 # Show that each of the following numbers is a perfect square.

Question:

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

Solution:

A perfect square is a product of two perfectly equal numbers.

(i) Resolving into prime factors:

$1225=25 \times 49=5 \times 5 \times 7 \times 7=5 \times 7 \times 5 \times 7=35 \times 35=(35)^{2}$

Thus, 1225 is the perfect square of 35.

(ii) Resolving into prime factors:

$2601=9 \times 289=3 \times 3 \times 17 \times 17=3 \times 17 \times 3 \times 17=51 \times 51=(51)^{2}$

Thus, 2601 is the perfect square of 51.

(iii) Resolving into prime factors:

$5929=11 \times 539=11 \times 7 \times 77=11 \times 7 \times 11 \times 7=77 \times 77=(77)^{2}$

Thus, 5929 is the perfect square of 77.

(iv) Resolving into prime factors:

$7056=12 \times 588=12 \times 7 \times 84=12 \times 7 \times 12 \times 7=(12 \times 7)^{2}=(84)^{2}$

Thus, 7056 is the perfect square of 84.

(v) Resolving into prime factors:

$8281=49 \times 169=7 \times 7 \times 13 \times 13=7 \times 13 \times 7 \times 13=(7 \times 13)^{2}=(91)^{2}$

Thus, 8281 is the perfect square of 91.