We know that (12 + 22 + 32 + ... + n2) = | 1 | n(n + 1)(2n + 1) |
6 |
Putting n = 10, required sum = | ❨ | 1 | x 10 x 11 x 21 | ❩ | = 385 |
6 |
Given Exp. = 35 + 15 x | 3 | = 35 + | 45 | = 35 + 22.5 = 57.5 |
2 | 2 |
∴ x = 5k + 3
⟹ x2 = (5k + 3)2
= (25k2 + 30k + 9)
= 5(5k2 + 6k + 1) + 4
∴On dividing x2 by 5, we get 4 as remainder.
Then, (5 + x + 2) must be divisible by 3. So, x = 2.
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.
(963 + 476)2 + (963 - 476)2 | =? |
(963 x 963 + 476 x 476) |
Given Exp. = | (a + b)2 + (a - b)2 | = | 2(a2 + b2) | = 2 |
(a2 + b2) | (a2 + b2) |
7429 Let 8597 - x = 3071 -4358 Then, x = 8597 - 3071 ---- = 5526 3071 ----
Then, x2 = (6q + 3)2
= 36q2 + 36q + 9
= 6(6q2 + 6q + 1) + 3
Thus, when x2 is divided by 6, then remainder = 3.
Given Exp. | = a2 + b2 - 2ab, where a = 287 and b = 269 |
= (a - b)2 = (287 - 269)2 | |
= (182) | |
= 324 |
Now, 1397 = 11 x 127
∴ The required 3-digit number is 127, the sum of whose digits is 10.
∴ (4x + 6y) = ( 4 x 5 + 6 x 1) = 26, which is not divisible by 11;
(x + y + 4 ) = (5 + 1 + 4) = 10, which is not divisible by 11;
(9x + 4y) = (9 x 5 + 4 x 1) = 49, which is not divisible by 11;
(4x - 9y) = (4 x 5 - 9 x 1) = 11, which is divisible by 11.
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