Numbers with Explanation

Q15. (5 + 14 + 23 + 32 + ... + 113) = ?

Answer- Option A

Explanation-

Given series is an AP in which a = 5, d = (14 - 5) = 9, and l = 113
Let the number of terms be n. Then,
a + (n - 1)d = 113
5 + (n - 1)x9 = 113
9n = 117
n = 13
Required sum =

n 2

(a + l) =

13 2

(5 + 113) = 767

Q16. What must least number be added from 2010 to obtain a number which is wholly divisible by 19?

	19 ) 2010 ( 105
	     19
	     -----
	       110
	        95
	       -----
	         15  
	       -----      

Q17. How many natural numbers between 11 and 99 are divisible by 7?

Answer- Option D

Explanation-

These numbers are 14, 21, 28, 75, ..., 98.
This is clearly an A.P. in which a = 14, d = (21 - 14) = 7 and l = 99.
Let the number of terms be n. Then,
98 = a + (n - 1)d
98 = 14 + (n - 1)x7
7n = 91
n = 13
Required numbers = 13

Q18. On dividing an individual number by 342, we get 47 as remainder. If the same number is divided by 18, what will be the remainder?

Q19. Find the sum (3 + 3² + 3³ + 3⁴ + ... + 3⁹)

Answer- Option B

Explanation-

This is GP in which a = 3, r =

32 3

= 3

Required sum =

a(rn - 1) (r - 1)

=

3(310 - 1) (3 - 1)

=

3x59048 2

= 88572

Q20. What is the sum of all even natural numbers less than 87?

Answer- Option C

Explanation-

Required sum = 2 + 4 + 6 + ... + 86
This is clearly an A.P. in which a = 2, d = 2, and l = 86
Clearly n = 43
Required sum =

n 2

(a + l) =

43 2

(2 + 86) = 1892

Q21. Simplify: = ?

Answer- Option A

Explanation-

Givn Exp. =

(658)3 - (328)3 (658)2 + 658 x 328 + 3282

=

(a)3 - (b)3 (a)2 + a x b + b2

, where a = 658 and b = 328

= (a - b) = (658 - 328) = 330.