Aptitude - Numbers

Let the two consecutive even integers be 2n and (2n + 2). Then,

(2n + 2)² = (2n + 2 + 2n)(2n + 2 - 2n)

     = 2(4n + 2)

     = 4(2n + 1), which is divisible by 4.

551 > 22

All prime numbers less than 24 are : 2, 3, 5, 7, 11, 13, 17, 19, 23.

119 is divisible by 7; 187 is divisible by 11; 247 is divisible by 13 and 551 is divisible by 19.

So, none of the given numbers is prime.

Required sum = (2 + 4 + 6 + ... + 30)

This is an A.P. in which a = 2, d = (4 - 2) = 2 and l = 30.

Let the number of terms be n. Then,

t_n = 30    a + (n - 1)d = 30

2 + (n - 1) x 2 = 30

n - 1 = 14

n = 15

Sn =	n	(a + l)	=	15	x (2 + 30)   = 240.
	2			2

99 = 11 x 9, where 11 and 9 are co-prime.

By hit and trial, we find that 114345 is divisibleby 11 as well as 9. So, it is divisible by 99.

This is an A.P. in which a = 6, d = 6 and S_n = 1800

Then,	n	[2a + (n - 1)d] = 1800
	2

	n	[2 x 6 + (n - 1) x 6] = 1800
	2

3n (n + 1) = 1800

n(n + 1) = 600

n² + n - 600 = 0

n² + 25n - 24n - 600 = 0

n(n + 25) - 24(n + 25) = 0

(n + 25)(n - 24) = 0

n = 24

Number of terms = 24.

This is an A.P. in which a = 51, l = 100 and n = 50.

Sum =	n	(a + l)	=	50	x (51 + 100)   = (25 x 151)   = 3775.
	2			2

Unit digit in 7⁹⁵ = Unit digit in [(7⁴)²³ x 7³]
= Unit digit in [(Unit digit in(2401))²³ x (343)]
= Unit digit in (1²³ x 343)
= Unit digit in (343)
= 3

Unit digit in 3⁵⁸ = Unit digit in [(3⁴)¹⁴ x 3²]
= Unit digit in [Unit digit in (81)¹⁴ x 3²]
= Unit digit in [(1)¹⁴ x 3²]
= Unit digit in (1 x 9)
= Unit digit in (9)
= 9

Unit digit in (7⁹⁵ - 3⁵⁸) = Unit digit in (343 - 9) = Unit digit in (334) = 4.

So, Option B is the answer.