Is 5 a divisor of 3^444 & 4^333?

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Without using a calculator, how can I know if 5 is a divisor of 3^444 and/or 4^333?
 
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Fermat's Little theorem, how else can you say anything about anything?
 
Did you learn about prime factorization in grade school? :biggrin:
 
true, i misread it.
 
Manchot said:
Did you learn about prime factorization in grade school? :biggrin:

No... :rolleyes:
 
Well, you should have! Any number can be written, in a unique way, as a product of primes. How would 3444 be written as a product of primes? Any 5s in there? (Hint: 3 and 5 are prime numbers.)

How would 4333 be written as a product of primes? (Hint: 4 is not a prime number but 2 is.)
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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