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) |
Here a = 3 and r = | 6 | = 2. Let the number of terms be n. |
3 |
Then, tn = 384 ⟹ arn-1 = 384
⟹ 3 x 2n - 1 = 384
⟹ 2n-1 = 128 = 27
⟹ n - 1 = 7
⟹ n = 8
∴ Number of terms = 8.
35423 317 x 89 = 317 x (90 -1 ) + 7164 = (317 x 90 - 317) + 41720 = (28530 - 317) ----- = 28213 84307 - 28213 ----- 56094 -----
This is an A.P. in which a = 10, d = 5 and l = 95.
tn = 95 ⟹ a + (n - 1)d = 95
⟹ 10 + (n - 1) x 5 = 95
⟹ (n - 1) x 5 = 85
⟹ (n - 1) = 17
⟹ n = 18
∴Requuired Sum = | n | (a + l) | = | 18 | x (10 + 95) = (9 x 105) = 945. |
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.
Given Exp. = 35 + 15 x | 3 | = 35 + | 45 | = 35 + 22.5 = 57.5 |
2 | 2 |
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 |
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.
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