For each of the following numbers find the smallest whole number by which it should be multiplied so as to get a perfect square also find the square root of the square number so obtained.
1. 252
2. 180
3. 1008
4. 2028
5.1358
6. 768
Answers
1) 252
→ 252 = 2 × 2 × 3 × 3 × 7
Here, 2 and 3 are in pair. But 7 is not in pair. So, it is not a perfect square.
Multiply with 7 on both sides
→ 252 × 7 = 2 × 2 × 3 × 3 × 7 × 7
→ 1764 = 2 × 2 × 3 × 3 × 7 × 7
√1764 = 42
7 is the smallest number by which 1764 multiples and becomes a perfect square.
____________________________
2) 180
→ 180 = 3 × 3 × 2 × 2 × 5
Here, 2 and 3 are in pair. But 5 is not in pair. So, it is not a perfect square.
Multiply with 5 on both sides
→ 180 × 5 = 3 × 3 × 2 × 2 × 5 × 5
→ 900 = 3 × 3 × 2 × 2 × 5 × 5
√900 = 30
5 is the smallest number by which 180 multiples and becomes a perfect square.
____________________________
3) 1008
→ 1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7
Here, 2 and 3 are in pair. But 7 is not in pair. So, it is not a perfect square.
Multiply with 7 on both sides
→ 1008 × 7 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
→ 7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
√7056 = 84
7 is the smallest number by which 180 multiples and becomes a perfect square.
____________________________
4) 2028
2028 = 2 × 2 × 3 × 13 × 13
Here, 2 and 13 are in pair. But 3 is not in pair. So, it is not a perfect square.
Multiply with 3 on both sides
→ 2028 × 3 = 2 × 2 × 3 × 13 × 13 × 3
→ 6084 = 2 × 2 × 3 × 13 × 13 × 3
√6084 = 78
3 is the smallest number by which 180 multiples and becomes a perfect square.
____________________________
5) 1358
→ 1358 = 2 ×
Well.. check it again the numbers are wrong.
___________________________
6) 768
→ 768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3
Here, 2 is in pair. But 3 is not in pair. So, it is not a perfect square.
Multiply with 3 on both sides
→ 768 × 3 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
→ 2304 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
√2304 = 48
3 is the smallest number by which 180 multiples and becomes a perfect square.
____________________________
(i) 252
Answer: By prime factorisation we get,
252 = 2 x 2 x 3 x 3 x 7
Here, 2 and 3 are in pairs but 7 needs a pair. Thus, 7 can become pair after multiplying 252 with 7.
So, 252 will become a perfect square when multiplied by 7.
Thus, Answer = 7
(ii) 180
Answer: By prime factorisation, we get, 180 = 3 x 3 x 2 x 2 x 5
Here, 3 and 2 are in pair but 5 needs a pair to make 180 a perfect square.
180 needs to be multiplied by 5 to become a perfect square.
Thus, Answer = 5
(iii) 1008
Answer: By prime factorisation of 1008, we get
1008 = 2 x 2 x 2 x 2 x 3 x 3 x 7
Here, 2 and 3 are in pair, but 7 needs a pair to make 1008 a perfect square.
Thus, 1008 needs to be multiplied by 7 to become a perfect square
Hence, Answer = 7
(iv) 2028
Answer: By prime factorisation of 2028, we get
2028 = 2 x 2 x 3 x 13 x 13
Here, 2 and 13 are in pair, but 3 needs a pair to make 2028 a perfect square.
Thus, 2028 needs to be multiplied by 3 to become a perfect square.
Hence, Answer = 3
(v) 1458
Answer: By prime factorisation of 1458, we get
1458 = 2 x 3 x 3 x 3 x 3 x 3 x 3
Here, 3 are in pair, but 2 needs a pair to make 1458 a perfect square.
So, 1458 needs to be multiplied by 2 to become a perfect square.
Therefore, Answer = 2
(vi) 768
Answer: By prime factorisation of 768, we get
768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3
Here, 2 are in pair, but 3 needs a pair to make 768 a perfect square.
So, 768 needs to be multiplied by 3 to become a perfect square.
Hence, Answer = 3