1) By what least number should 2028 be multiplied so that the product is a perfect square? Find the square root of the product so obtained
2). By what least number should 10368 be divided so that the quotient is a perfect square? Find the square root of the quotient so obtained.
3) What least number must be added to 368448 to make the sum a perfect square? Find this perfect square and its square root.
4) What least number must be added to 93020 to make the sum a perfect square? Find this perfect square and its square root.
Answers
Step-by-step explanation:
Ans. 1) Thus, 2028 needs to be multiplied by 3 to become a perfect square. Therefore, the number 6084 has 3 pairs of equal prime factors . Hence, the smallest number by which 2028 must be multiplied so that the product is a perfect square is 7. And the square root of the new number is √6084=78.
Ans. 2) 10368=2×2×2×2×2×2×2×3×3×3×3
On grouping the prime factors of 10368; one factor i.e. 2 is left which cannot be paired with equal factor.
Taking L.C.M.
∴ The given number should be divided by 2
Now
2
10368
=
2
2×2×2×2×2×2×2×3×3×3×3
(Splitting the terms)
=2×2×2×3×3=72
Ans. 3) The following steps to find the square root by long division method
1. Draw lines over pairs of digits from right to left.
2. Find the greatest number whose square is less than or equal to the digits in the first group.
3. Take this number as the divisor and quotient of the first group and find the remainder.
4. Move the digits from the second group besides the remainder to get the new dividend.
5. Double the first divisor and bring it down as the new divisor.
6. Complete the divisor and continue the division.
7. Repeat the process till the remainder becomes zero
Divisor
↓
Quotient
↓
26
2
7
00
4
46
300
276
24← Remainder
We observe here 26
2
<700<27
2
The required number to be added = 27
2
−700
= 729 - 700
= 29
Therefore, 29 must be added to 700 to make it a perfect square.
Ans. 4). Don't know