Exercise 6.1

1. What will be the units digit of the square of the following numbers?

(i) 39

Solution:

The units digit of 39 is .

The units digit of 92 (9×9 = ) is .

Therefore, the units digit of 392 will be .

(ii) 297

Solution:

The units digit of 297 is .

The units digit of 72 (7×7 = ) is .

Therefore, the units digit of 2972 will be .

(iii) 5125

Solution:

The units digit of 5125 is .

The units digit of 52 (5×5 = ) is .

Therefore, the units digit of 51252 will be .

(iv) 7286

Solution:

The units digit of 7286 is .

The units digit of 62 (6×6 = ) is .

Therefore, the units digit of 72862 will be .

(v) 8742

Solution:

The units digit of 8742 is .

The units digit of 22 (2×2 = ) is .

Therefore, the units digit of 87422 will be .

2. Which of the following numbers are perfect squares?

(i) 121

Solution:

121 is a perfect square: YesNo

121 = × = 112122.

Therefore, 121 is a perfect square.

(ii) 136

Solution:

136 is a perfect square: NoYeso

136 cannot be expressed as the square of an integer.

Therefore, 136 is not a perfect square.

(iii) 256

Solution:

256 is a perfect square: YesNo

256 = × = 162152

Therefore, 256 is a perfect square.

(iv) 321

Solution:

321 is a perfect square: NoYes

321 be expressed as the square of an integer.

Therefore, 321 is not a perfect square.

(v) 600

Solution:

600 is a perfect square: NoYes

600 be expressed as the square of an integer.

Therefore, 600 is not a perfect square.

3. The following numbers are not perfect squares. Give reasons?

(i) 257

Solution:

257 is a perfect square: NoYes

257 ends in .

Perfect squares can only end in 0, 1, 4, 5, 6, or 9.

Therefore, 257 is not a perfect square.

(ii) 4592

Solution:

4592 is a perfect square: NoYes

4592 ends in .

Therefore, 4592 is not a perfect square.

(iii) 2433

Solution:

2433 is a perfect square: NoYes

2433 ends in .

Therefore, 2433 is not a perfect square.

(iv) 5050

Solution:

5050 is a perfect square: NoYes

5050 ends in , but it has an number of zeros.

Therefore, 5050 is not a perfect square.

(v) 6098

Solution:

6098 is a perfect square: NoYes

6098 ends in .

Therefore, 6098 is not a perfect square.

4. Find whether the square of the following numbers are even or odd?

(i) 431

Solution:

431 is evenodd.

The square of an odd number is always .

Therefore, 4312 is odd.

(ii) 2826

Solution:

2826 is evenodd.

The square of an even number is always .

Therefore, 28262 is even.

(iii) 8204

Solution:

8204 is evenodd.

The square of an even number is always .

Therefore, 82042 is even.

(iv) 17779

Solution:

17779 is evenodd.

The square of an odd number is always .

Therefore, 177792 is odd.

(v) 99998

Solution:

99998 is evenodd.

The square of an even number is always .

Therefore, 999982 is even.

5. How many numbers lie between the square of the following numbers?

(i) 25; 26

Solution:

252 =

262 =

The numbers between 252 and 262 are 626, 627, ..., 675.

Number of such numbers = - + 1 = .

(ii) 56; 57

Solution:

562 =

572 =

The numbers between 562 and 572 are 3137, 3138, ..., 3248.

Number of such numbers = - + 1 = .

(iii) 107; 108

Solution:

1072 =

1082 =

The numbers between 1072 and 1082 are 11450, 11451, ..., 11663.

Number of such numbers = - + 1 = .

6. Without adding, find the sum of the following numbers

(i) 1 + 3 + 5 + 7 + 9 =

Solution:

This is the sum of the first 5 odd numbers.

The sum of the first 'n' odd numbers is n22(n+1).

Therefore, the sum is 52 = .

(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 =

Solution:

This is the sum of the first odd numbers.

The sum of the first 'n' odd numbers is n22(n+1).

Therefore, the sum is 92 = .

(iii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 =

Solution:

This is the sum of the first odd numbers.

The sum of the first 'n' odd numbers is n22(n+1).

Therefore, the sum is 132 = .