Sum of digits = (5 + 1 + 7 + x + 3 + 2 + 4) = (22 + x), which must be divisible by 3.
x = 2.
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.
(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.
Let Sn =(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:
Sn = | n(n + 1) | = | 45(45 + 1) | = 1035. |
2 | 2 |
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.
753 x 753 + 247 x 247 - 753 x 247 | = ? |
753 x 753 x 753 + 247 x 247 x 247 |
x + 3699 + 1985 - 2047 = 31111
x + 3699 + 1985 = 31111 + 2047
x + 5684 = 33158
x = 33158 - 5684 = 27474.
Sum of digits = (4 + 8 + 1 + x + 6 + 7 + 3) = (29 + x), which must be divisible by 9.
x = 7.