The remainder when 1992 is divided by 92 is
a) 41
b) 49
c)32
d)44
1The idea is that , 19 f ( p ) = 1 ( mod 92 ) , where f ( p ) represents Euler's totient function , i . e , it caculates the number of co - prime numbers less than or equal to p . Its value is , in this case ---
f ( P ) = 92 ( 1 - 1 / 2 ) ( 1 - 1 / 23 ) = 44
So , 19 44 = 1 ( mod 92 )
or , 1 9 88 = 1 ( mod 92 )
So , 19 92 = 19 4 ( mod 92 ) = 361 2 ( mod 92 ) = ( - 7 ) 2 ( mod 92 ) = 49 ( mod 92 )
1Chinese Remainder theorem (along with other results).
First note 92= 4 × 23 with gcd(4,23) =1.
Let us call N= 1992.
We will compute, N(mod 4) and N (mod 23) and then use CRT tocompute N (mod 92).
First, N (mod 4) = (19)92( ð‘šð‘œð‘‘ 4) = (−1)92(ð‘šð‘œð‘‘ 4) = 1and ð‘(ð‘šð‘œð‘‘ 23) = 194.[(19)22 (ð‘šð‘œð‘‘ 23)]2(ð‘šð‘œð‘‘ 23) = (−4)4(ð‘šð‘œð‘‘ 23) = (16)4(ð‘šð‘œð‘‘ 23) = (−7)2(ð‘šð‘œð‘‘ 23) = 49(ð‘šð‘œð‘‘ 23) = 3.
Note in the above we have used Fermat’s Little Theorem.
Now, If you know CRT,
you candirectly say ð‘( ð‘šð‘œð‘‘ 92) = 49.
If not, you can compute it.