Question

what is the unit digit of 4^5^6^7^8^9 ?

Answer 1

It would be surely zero or 5 as when u multiply by 5 u get 5 0r 0 and as I did it in calculator I got 0

Answer 2

The answer is 6. 

The possible unit digits of powers of 4 are 4 and 6. 

4 → 4 x 4 = 16 → 6 x 4 = 24 

Here, only 4 and 6 come in unit place, i. e., only two digits. 

If we get 1 as remainder on dividing the exponent of 4 by 2, the unit digit of power of 4 must be 4, and if 0 is gotten as remainder, the unit digit must be 6. 

Here, on $4^5$, 1 is gotten as remainder on dividing 5 by 2. So the unit digit of $4^5$ must be 4. ($4^5 = 102<strong>4</strong>$) 

Let's check $4^6$ (The unit digit of $4^5$ ^ next exponent in the question). 

6 ÷ 2 gives remainder 0. So the unit digit of $4^6$ is 6. ($4^6 = 409<strong>6</strong>$) 

Here, 6 comes as unit digit. 

The possible unit digit of powers of 6 is only 6. 

6 → 6 x 6 = 36 

Here, only 6 comes in unit place. 

So the unit digit of powers of 6 is always 6. 

∴ $6^7$ → unit digit 6. 

$6^8$ → unit digit 6. 

$6^9$ → unit digit 6. 

∴ $4^{5^{6^{7^{8^{9}}}}}$ → unit digit 6. 

Hope this will be helpful.