A number theory problem by Wilson Widyadhana

Number Theory Level 2

$\Large 2^{201620152014} \bmod 5 = \, ?$

The answer is 4.

2 solutions

Pham Khanh
May 8, 2016

$\large 2^{201620152014}=4^{100810076007} \equiv (-1)^{100810076007} \pmod 5 \qquad (*)$ Cause $100810076007$ is odd so $\large (*) \iff 2^{201620152014} \equiv (-1)^{100810076007} \equiv -1 \equiv \boxed{4} \pmod 5$

Wilson Widyadhana
May 7, 2016

Since $2^2 = 4 (mod \ 5)$ , in which 2 is a factor of 201620152014, $\Large 2^{201620152014} = \ \boxed{4} (mod \ 5) _\square$

Moderator note:

This solution is incomplete, as it hasn't provided a sufficient argument. For example, 2 is a factor of 4, but $2^4 \not \equiv 4 \pmod{5}$ .

But if you says so, this could only mean to $\Large 2^{201620152014}=4^{100810076007}$ You haven't says that why $\Large 4^{100810076007} \equiv 4 \pmod{5}$

Pham Khanh - 5 years, 1 month ago

A number theory problem by Wilson Widyadhana

The answer is 4.

2 solutions

Moderator note:

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