Figure It Out 1.1

Question 1

Which of the following numbers are not perfect squares? (i) 2032 (ii) 2048 (iii) 1027 (iv) 1089

Solution: We check the unit digits. A perfect square cannot end in 2, 3, 7, or 8.

• (i) 2032 ends in 2. Not a square.
• (ii) 2048 ends in 8. Not a square.
• (iii) 1027 ends in 7. Not a square.
• (iv) 1089 ends in 9. It might be a square ($33^2 = 1089$).

Answer: (i), (ii), and (iii) are not perfect squares.

Question 2

Which one among $64^2, 108^2, 292^2, 36^2$ has last digit 4?

Solution: The last digit of a square depends on the last digit of the number.

• $64^2 \to 4 \times 4 = 16$ (ends in 6)
• $108^2 \to 8 \times 8 = 64$ (ends in 4)
• $292^2 \to 2 \times 2 = 4$ (ends in 4)
• $36^2 \to 6 \times 6 = 36$ (ends in 6)

Answer: $108^2$ and $292^2$.

Question 3

Given $125^2 = 15625$, what is the value of $126^2$?

Solution: We can use the identity $(n+1)^2 = n^2 + (n) + (n+1) = n^2 + 2n + 1$. Here $n = 125$. $126^2 = 125^2 + 125 + 126$ $126^2 = 15625 + 251$

Answer: (iv) $15625 + 251$.

Question 4

Find the length of the side of a square whose area is 441 m².

Solution: Area = Side $\times$ Side = $x^2 = 441$. We need to find $\sqrt{441}$. Prime factors of 441: $441 = 3 \times 147 = 3 \times 3 \times 49 = 3 \times 3 \times 7 \times 7$. Group pairs: $(3 \times 3) \times (7 \times 7)$. $\sqrt{441} = 3 \times 7 = 21$.

Answer: 21 m.

Question 5

Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.

Solution:

1. Find the LCM of 4, 9, 10.
   • $4 = 2^2$
   • $9 = 3^2$
   • $10 = 2 \times 5$
   • LCM = $2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$.
2. Check prime factors of 180: $180 = 2 \times 2 \times 3 \times 3 \times 5$.
3. To make it a perfect square, all factors must be in pairs.
4. The factor 5 is unpaired. We must multiply by 5.
5. $180 \times 5 = 900$.

Answer: 900.

Question 6

Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.

Solution:

1. Prime factorization of 9408: $9408 \div 2 = 4704$ $4704 \div 2 = 2352$ $2352 \div 2 = 1176$ $1176 \div 2 = 588$ $588 \div 2 = 294$ $294 \div 2 = 147$ $147 \div 3 = 49$ $49 \div 7 = 7$ $7 \div 7 = 1$ Factors: $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 7 \times 7$.
2. Group pairs: $(2\times2), (2\times2), (2\times2), (7\times7)$.
3. The factor 3 is left unpaired.
4. Smallest number to multiply is 3.
5. New number: $9408 \times 3 = 28224$.
6. Square root = $2 \times 2 \times 2 \times 7 \times 3 = 8 \times 21 = 168$.

Answer: Multiply by 3. Square root of product is 168.

Question 7

How many numbers lie between the squares of the following numbers? (i) 16 and 17 (ii) 99 and 100

Solution: Between $n^2$ and $(n+1)^2$, there are $2n$ non-square numbers. (i) Here $n=16$. Numbers = $2 \times 16 = 32$. (ii) Here $n=99$. Numbers = $2 \times 99 = 198$.

Question 8

Fill in the missing numbers: $4^2 + 5^2 + 20^2 = (\underline{\quad})^2$ $9^2 + 10^2 + (\underline{\quad})^2 = (\underline{\quad})^2$

Solution: Pattern: $n^2 + (n+1)^2 + [n(n+1)]^2 = [n(n+1)+1]^2$.

1. $4^2 + 5^2 + 20^2$. Here $4 \times 5 = 20$. Next is 21. Answer: 21
2. $9^2 + 10^2 + (90)^2$. Next is 91. Answer: 90, 91