Euler S Theorem

To find the product of the last two digits of , we first determine the last two digits of the number, which is equivalent to finding . We can use Euler's theorem to simplify the calculation.

Euler's theorem states that for any integer that is coprime to , , where is Euler's totient function. For , we determine:

.

Thus, . Since , our task simplifies to finding .

Now, calculate to use the reduced exponent. Since , we have:

.

Thus, we need to find .

Using successive squaring:

Now, break down :

.

Calculate using:

.

Calculate individually:

So, .

Thus, .

The last two digits of are therefore 79. The product of these digits is:

.

This value falls within the given range (63,63).