12345679 x 72 | = 12345679 x (70 +2) |
= 12345679 x 70 + 12345679 x 2 | |
= 864197530 + 24691358 | |
= 888888888 |
2133 → 9 (X)
2343 → 12 (/)
3474 → 18 (X)
4131 → 9 (X)
5286 → 21 (/)
5340 → 12 (/)
6336 → 18 (X)
7347 → 21 (/)
8115 → 15 (/)
9276 → 24 (/)
Required number of numbers = 6.
8796 x 223 + 8796 x 77 | = 8796 x (223 + 77) [Ref: By Distributive Law ] |
= (8796 x 300) | |
= 2638800 |
= (22 x 12) + (22 x 22) + (22 x 32) + ... + (22 x 102)
= 22 x [12 + 22 + 32 + ... + 102]
[ | Ref: (12 + 22 + 32 + ... + n2) = | 1 | n(n + 1)(2n + 1) | ] | |
6 |
= | ❨ | 4 x | 1 | x 10 x 11 x 21 | ❩ |
6 |
= (4 x 5 x 77)
= 1540.
= Unit digit in [(42)896 x 4]
= Unit digit in (6 x 4) = 4
Unit digit in (625)317 = Unit digit in (5)317 = 5
Unit digit in (341)491 = Unit digit in (1)491 = 1
Required digit = Unit digit in (4 x 5 x 1) = 0.
(6 + 5 + 2 + 9) - (x + 1 + 7) = (14 - x), which must be divisible by 11.
∴ x = 3
6) 4456 (742 42 --- 25 24 Therefore, Required number = (6 - 4) = 2. --- 16 12 --- 4
[ | Ref: (12 + 22 + 32 + ... + n2) = | 1 | n(n + 1)(2n + 1) | ] | |
6 |
= | ❨ | 20 x 21 x 41 | - | 10 x 11 x 21 | ❩ |
6 | 6 |
= (2870 - 385)
= 2485.
(17200 - 1200) is completely divisible by (17 + 1), i.e., 18.
⟹ (17200 - 1) is completely divisible by 18.
⟹ On dividing 17200 by 18, we get 1 as remainder.
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