What will be remainder when (67⁶⁷ + 67) is divided by 68?

Correct Answer: 80578

Explanation:

(217)2 + (183)2	= (200 + 17)2 + (200 - 17)2
	= 2 x [(200)2 + (17)2]       [Ref: (a + b)2 + (a - b)2 = 2(a2 + b2)]
	= 2[40000 + 289]
	= 2 x 40289
	= 80578.

Correct Answer: 1022

Explanation:

This is a G.P. in which a = 2, r =	22	= 2 and n = 9.
	2

∴Sn =	a(rn - 1)	=	2 x (29 - 1)	= 2 x (512 - 1)   = 2 x 511   = 1022.
	(r - 1)		(2 - 1)

Correct Answer: 3625216

Explanation:

1904 x 1904	= (1904)2
	= (1900 + 4)2
	= (1900)2 + (4)2 + (2 x 1900 x 4)
	= 3610000 + 16 + 15200.
	= 3625216.

Correct Answer: 6

Explanation:

72 = 9 x8, where 9 and 8 are co-prime.

The minimum value of x for which 73x for which 73x is divisible by 8 is, x = 6.

Sum of digits in 425736 = (4 + 2 + 5 + 7 + 3 + 6) = 27, which is divisible by 9.

∴Required value of * is 6.

Correct Answer: 251001

Explanation:

Given Exp.	= (397)2 + (104)2 + 2 x 397 x 104
	= (397 + 104)2
	= (501)2 = (500 + 1)2
	= (5002) + (1)2 + (2 x 500 x 1)
	= 250000 + 1 + 1000
	= 251001

Correct Answer: 7+⁸¹⁷⁄₁₀₂₀

Explanation:

Given sum	= 9 + 3 + 7 + 2 - ❨ 9 + 1 ❩ 4 17 15
	= (9 + 7 - 9) + ❨ 3 + 2 - 1 ❩ 4 17 15
	= 7 + 765 + 120 - 68 1020
	= 7 + 817 1020

Correct Answer: 415624

Explanation:

(4 + 5 + 2) - (1 + 6 + 3) = 1, not divisible by 11.

(2 + 6 + 4) - (4 + 5 + 2) = 1, not divisible by 11.

(4 + 6 + 1) - (2 + 5 + 3) = 1, not divisible by 11.

(4 + 6 + 1) - (2 + 5 + 4) = 0, So, 415624 is divisible by 11.

Correct Answer: 3893103

Explanation:

3897 x 999	= 3897 x (1000 - 1)
	= 3897 x 1000 - 3897 x 1
	= 3897000 - 3897
	= 3893103.

Correct Answer: None of these

Explanation:

Given Exp. =	800	x	1296	= 450
	64		36

Correct Answer: 2029272

Explanation:

2056 x 987	= 2056 x (1000 - 13)
	= 2056 x 1000 - 2056 x 13
	= 2056000 - 26728
	= 2029272.