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Fermat's Little Theorem : Maths Trick 21

Maths-Trick-21

This trick is on Number Theory & it is very useful for any entrance examinations.


Fermat’s Little Theorem: If p is a prime & p is not a divisor of a, then
                                            ap-1≡1(modp)
                                   ap ≡ a (modp) ak(p - 1)+1 ≡ a (modp) ,    where p is a prime & k,a .  For proof see any standard book.

Q. [TIFR’2012] T/F : If n is not a multiple of 23 then the remainder when n11 is divided by 23 is  ± 1 (mod23).

Solution :- TRUE .
Using Fermat’s Theorem:  n22 ≡ 1 ( mod23 )
                                    n11 ± 1 ( mod23 ).

Q. [TIFR’2011] The equation  x22 ≡ 2 ( mod23 ) has
(a) No solution    (b) 23 solutions    (c) exactly one solution   (d) 22 solutions

Solution :- (a)  By Fermat’s Little Theorem:  x23-1 ≡ 1 ( mod23 )
                                                             x22 ≡ 1 ( mod23 ) .