number theory

Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?

Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$ ?

The value of 3 51 mod 5 is_________.

X is a 30 digit number starting with the digit 4 followed by the digit 7. Then the number X³ will have

The value of the expression $${13^{99}}$$ ($$mod$$ $$17$$), in the range $$0$$ to $$16,$$ is ______________ .

The number of divisors of $$2100$$ is ___________.

The number of distinct positive integral factors of 2014 is _______ .

What is the probability that divisor of $${10^{99}}$$ is a multiple of $${10^{96}}$$ ?

The exponent of $$11$$ in the prime factorization of $$300!$$ is

Let $$n = {p^2}q,$$ where $$p$$ and $$q$$ are distinct prime numbers. How many numbers $$m$$ satisfy $$1 \le m \le n$$ and $$gcd\left( {m.n} \right) = 1?$$ Note that $$gcd(m,n)$$ i...