Number System — CDS Elementary Mathematics PYQs

What is the sum of all 3-digit numbers that give a remainder of 5 when they are divided by 50?

Let $p$ be the remainder when $7^{84}$ is divided by 342 and $q$ be the remainder when $7^{84}$ is divided by 344. What is $(p-q)$ equal to?

$N$ is the smallest 5-digit number which when divided by $2,2^2,2^3,2^4,\dots ,2^n$ leaves a remainder 1. What is the value of $n$?

What is the minimum value of $p$ for which $\frac{1}{532900}+\frac{p^2}{266450}+\frac{p^4}{523900}$ is an integer?

Consider a 2-digit number $N$. Let $P$ be the product of the digits of the number. If $P$ is added to the square of the digit in the tens place of $N$, we get 84. If $P$ is added to the square of the digit in the unit place of $N$, we get 60. What is the value of $P+N$?

Given that $\frac{100\times 99\times 98\times \cdots \times 3\times 2\times 1}{100^n}$ is an integer. What is the largest value of $n$ for which this is true?

There are two natural numbers $m$ and $n$ $(m>n)$. When $m$ is divided by 12, it leaves a remainder 4. When $n$ is divided by 12, it leaves a remainder 6. Which of the following statements is/are correct?
I. The remainder when $(m+n)$ is divided by 12 is 10.
II. The remainder when $(m-n)$ is divided by 12 is 10.
Select the correct answer using the code given below:

What is the remainder when $2^p-1$ is divided by $p$, where $p>5$ is a prime number?

What is the number of factors of $24^3-16^3-8^3$?

If $p=0.\overline{09}$, then what is the value of $70p^2+43p-5$?