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Exponential remainder Common Admission Test (CAT) Problems

Basic Rules
- Remainder of a product = product of remainder of factors
   Eg. Remainder of 49/5 = Remainder of 7/5 x Remainder of 7/5
                                         = 2x2 = 4
- Remainder of a sum = sum of reminder of each term

 Expansion by binomial series
What is the remainder when 17^23 is divided by 16?
1723 = (16+1)23
= 23C0162310 + 23C1162211 +...+ 23C22161122 + 23C23160123
There are 24 terms with the first 23 containing a multiple of 16 and they are divisible by 16. The last term is 1 and that is the remainder when 1723 is divided by 16.

 What is the remainder when 20^20 is divided by 23?
2020 = (23-3)20
= 20C0232030 - 20C1231931 +...- 20C19231319 + 20C20230320
The remainder is the remainder of last term divided by 23
= remainder of 320/23
= remainder of (35)4 /23
= remainder of 2434/23
= remainder of (230+13)4/23
= remainder of 134 /23
= remainder of 1692/23
= remainder of (23x7+8)2/23
= remainder of 82/23
= remainder of 64/23
= remainder of (2x23+18)/23
= 18

What is the remainder when 43^86 is divided by 5?
4386 = (40+3)86
The remainder is the last of the binomial expression
= remainder of 386/5
= remainder of 943/5
= remainder of (5+4)43/5
= remainder of 443/5
= remainder of (1621 x 4)/5
= remainder of ((15+1)21 x 4)/5
= remainder of (121 x 4)/5
= 4

What is the remainder when 25^25 is divided by 9?
2525 = (18+7)25
The remainder is the last term of the binomial expression
= remainder of 725/9
= remainder of (4912 x 7)/9
= remainder of ((45+4)12 x 7)/9
= remainder of (412 x 7)/9
= remainder of (166 x 7)/9
= remainder of ((9+7)6 x 7)/9
= remainder of (76 x 7)/9
= remainder of (493 x 7)/9
= remainder of ((45+4)3 x 7)/9
= remainder of (43 x 7)/9
= remainder of (16 x 4 x 7)
= remainder of (7 x 4 x 7)
= remainder of (28 x 7)
= remainder of (1 x 7)
= 7

What is the remainder when 22006 is divided by 17?
22006 = (25)401 x 2
The remainder is the remainder of (32401 x 2)/17
= remainder of ((17+15)401 x 2)/17
= remainder of (15401 x 2)/17
= remainder of (225200 x 15 x 2)/17
= remainder of ((13x7+4)200 x 30)/17
= remainder of ((13x7+4)200 x 13)/17
= remainder of (4200 x 13)/17
= remainder of (2400 x 13)/17
= remainder of (3280 x 13)/17
= remainder of ((17+15)80 x 13)/17
= remainder of (1580 x 13)/17
= remainder of (22540 x 13)/17
= remainder of ((13x7+4)40 x 13)/17
= remainder of (440 x 13)/17
= remainder of (280 x 13)/17
= remainder of (3216 x 13)/17
= remainder of ((17+15)16 x 13)/17
= remainder of (1516 x 13)/17
= remainder of (2258 x 13)/17
= remainder of ((13x7+4)8 x 13)/17
= remainder of (48 x 13)/17
= remainder of (216 x 13)/17
= remainder of (323 x 2 x 13)/17
= remainder of ((17+15)3 x 26)/17
= remainder of ((17+15)3 x 9)/17
= remainder of (153 x 9)/17
= remainder of (225 x 15 x 9)/17
= remainder of ((13 x 7+4) x 15 x 9)/17
= remainder of (4 x 15 x 9)/17
= remainder of (60 x 9)/17
= remainder of ((17 x 3 + 9) x 9)/17
= remainder of (9 x 9)/17
= remainder of 81/17
= remainder of (17 x 4 + 13)/17
= 13

Cyclic Remainders
What is the remainder when 7^700 is divided by 100
The last 2 digits of 71 is 07, ie when 71 is divided by 100, 07 is the remainder
The last 2 digits of 72 is 49
The last 2 digits of 73 is 43 (Multiply 49 by 7 and retain the last 2 digits)
The last 2 digits of 74 is 01
The last 2 digits of 75 is 07
The last 2 digits of 76 is 49
The last 2 digits of 77 is 43
The last 2 digits of 78 is 01
The last 2 digits of 79 is 07

In general, The last 2 digits of 74n is 01, where n is any integer.
700 = 4*175 and hence when 7700 is divided by 100, 1 is the remainder.

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