Numbers - General Questions (15)

x + 3699 + 1985 - 2047 = 31111

   x + 3699 + 1985 = 31111 + 2047

   x + 5684 = 33158

   x = 33158 - 5684 = 27474.

Given Exp. =(a² + b² - ab)=1=1=1(a³ + b³)(a + b)(753 + 247)1000

24 = 3 x8, where 3 and 8 co-prime.

Clearly, 35718 is not divisible by 8, as 718 is not divisible by 8.

Similarly, 63810 is not divisible by 8 and 537804 is not divisible by 8.

Consider option (D),

Sum of digits = (3 + 1 + 2 + 5 + 7 + 3 + 6) = 27, which is divisible by 3.

Also, 736 is divisible by 8.

 3125736 is divisible by (3 x 8), i.e., 24.

Let S_n =(1 + 2 + 3 + ... + 45). This is an A.P. in which a =1, d =1, n = 45.

Sn =	n	[2a + (n - 1)d]	=	45	x [2 x 1 + (45 - 1) x 1]	=		45	x 46		= (45 x 23)
	2			2				2

= 45 x (20 + 3)

= 45 x 20 + 45 x 3

= 900 + 135

= 1035.

Shorcut Method:

19657         Let x - 53651  = 9999
 33994         Then, x = 9999 + 53651 = 63650
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 53651
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(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.

The smallest 3-digit number is 100, which is divisible by 2.

 100 is not a prime number.

101 < 11 and 101 is not divisible by any of the prime numbers 2, 3, 5, 7, 11.

√101 is a prime number.

Hence 101 is the smallest 3-digit prime number.

Sum of digits = (5 + 1 + 7 + x + 3 + 2 + 4) = (22 + x), which must be divisible by 3.

  x = 2.