Correct Quotient = 420 ? 21 = 20
All prime numbers less than 22 are : 2, 3, 5, 7, 11, 13, 17, 19.
161 is divisible by 7, and 221 is divisible by 13.
373 is not divisible by any of the above prime numbers.
∴373 is prime.
Largest 4-digit number = 9999 88) 9999 (113 88 ---- 119 88 ---- 319 264 --- 55 --- Required number = (9999 - 55) = 9944.
The smallest 6-digit number 100000. 111) 100000 (900 999 ----- 100 --- Required number = 100000 + (111 - 100) = 100011.
This is an A.P. in which a = 102, d = 6 and l = 996
Let the number of terms be n. Then,
a + (n - 1)d = 996
⟹ 102 + (n - 1) x 6 = 996
⟹ 6 x (n - 1) = 894
⟹ (n - 1) = 149
⟹ n = 150.
∴ Unit digit in 365 = Unit digit in [ (34)16 x 3 ] = (1 x 3) = 3
Unit digit in 659 = 6
Unit digit in 74 ⟹ Unit digit in (74)17 is 1.
Unit digit in 771 = Unit digit in [(74)17 x 73] = (1 x 3) = 3
∴ Required digit = Unit digit in (3 x 6 x 3) = 4.
x = 357 x y + 39
= (17 x 21 x y) + (17 x 2) + 5
= 17 x (21y + 2) + 5)
∴Required remainder = 5.
∴ 10 x Quotient = 230 ⟹ = | 230 | = 23 |
10 |
Dividend = (Divisor x Quotient) + Remainder
= (230 x 23) + 46
= 5290 + 46
= 5336.
But, unit digit in (74)26 = 1
∴ Unit digit in 7105 = (1 x 7) = 7
Given Exp. = | (1000)9 | = | (103)9 | = | (10)27 | = 10(27-24) = 103 = 1000 |
1024 | 1024 | 1024 |
106 x 106 - 94 x 94 | = (106)2 - (94)2 |
= (106 + 94)(106 - 94) [Ref: (a2 - b2) = (a + b)(a - b) ] | |
= (200 x 12) | |
= 2400. |
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