Arun Sharma Solutions for Exercise 3: Review Test 3

$\mathrm{N}={99}^3-{36}^3-{63}^3$, how many factors does $\mathrm{N}$ have?

Find the highest power of$2$ in $10!+11!+12!+13!+....98!+99!+100!.$

$100!$ is divisible by ${160}^n$, What is the maximum integral value $n$?

What is the remainder when $1\times 1+11\times 11+111\times 111+1111\times 1111+......................\left(2001 times 1\right)\times \left(2001 times 1\right)$is divided by$100$?

Find the sum of the first$125$ terms of the sequence $1, 2, 1, 3, 2, 1, 5, 4, 3, 2....$

Mr. Ramlal lived his entire life during the $1800s$. In the last year of his life, Ramlal stated: Once I was $x$year old in the year $x^2$. He was born in the year?

Find the remainder when $971\left({30}^{99}+{61}^{100}\right)\times (1148{)}^{56}$ is divided by $31$

Find the remainder when ${\left({10}^3+9^3\right)}^{1000}$ is divided by ${12}^3.$