Question

Find the remainder when 17³⁶ is divided by 36?

• A. 1
• B. -1
• C. 17
• D. 2

Answer 1

The correct option is A 1

Answer=b

Approach 1

Go by frequency method

$\frac{17}{36}$ gives a remainder $=17$

$\frac{{17}^2}{36}$ gives a remainder $=1$

Frequency $=2$

Odd powers remainder $=17$, even powers $=1$

Approach 2

Euler’s number of $36$ is $36\times \frac{1}{2}\times \frac{2}{3}=12$

$36=12k.$ hence, from Fermat theorem $\frac{{17}^{36}}{36}{|}_R=1$

Note :
Euler's number is the number of co_primes of number which is less than that number.
A number N can be written as $a^m{b}^n$ (where a and b are the prime factors of N)
eg) $20=2^2\times 5^1$
Here $a=2$ and $b=5,m=2$ and $n=1$
Euler's number $=N[1-\left(\frac{1}{a}\right)][1-\left(\frac{1}{b}\right)]$
From fermat Theorem
$\frac{N^{\left(Eule{r}^{\prime }snumberof{y}^{+}k\right)}}{y}{\mid }_R=1$ (i.e When a number N is raised to the Euler's number of a number)
$"y"$ is divided by $"y"$, the remainder will be 1