Question

The remainder when \[{75^{{{75}^{75}}}}\]is divided by 37.
A. 0
B. 1
C. 3
D. Can’t be determine

Answer 1

Hint: The given problem is the type of modulus operator. The modulus operator % can be used with integer type operands and always has integer type results. It is the integer type remainder of an integer division.

Complete Step-by-Step solution:
Given,
\[75{\text{ }} \cong \;1\;modulo{\text{ }}37\]
\[75{\text{ }} \times {\text{ }}75{\text{ }} \cong {\text{ }}1{\text{ }} \times 1{\text{ }}( = 1){\text{ }}modulo{\text{ }}37\]
$75 \times 75 \times 75 \cong 1 \times 1 \times 1( = 1)modulo37$
So,\[{75^{(any\operatorname{int} eger)}} \cong \infty 1modulo37\]
\[So,{75^{({{75}^7})}} \cong 1modulo37\]
Hence, the remainder is 1.

Note: The modulo operation is a way to determine the remainder of a division operation. Instead of returning the result of the division, the modulo operation returns the whole number remainder. That remainder is what the modulo operation returns. These types of problems need to be solved by specific methods as general methods will take a long time to solve.