Number System — CDS Elementary Mathematics PYQs

Let $P=5^5\times 15^{15}\times 25^{25}\times 35^{35}$ and $Q=10^{10}\times 20^{20}\times 30^{30}\times 40^{40}$. What is the number of consecutive zeros at the end of the sum $(P+Q)$?

Seven times a two-digit positive number is equal to four times the number obtained by reversing the order of the digits. How many such two-digit numbers are there?

$27^{27}-9^{40}-3^{79}$ is divisible by how many natural numbers less than 10?

Let $N$ be the greatest number that will divide 600, 631 and 724, leaving the same remainder. What is the value of $N$?

What is the sum of all the divisors of 256?

Consider the following statements:
I. Two consecutive natural numbers are always co-prime.
II. If $m$ and $n$ are relatively prime, then $m\times n$ is always even.

Which of the statements given above is/are correct?

What is the digit in the unit place of the number $1^1\times 3^3\times 5^5\times 7^7\times 9^9$?

Let $P=N^2$ where $N$ is an odd integer. What is the remainder when $P$ is divided by 8?

A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Is $(p^2+q^2)$ always a composite number, where $p$ and $q$ are different prime numbers?
Statement-I: $(p-q)$ is an odd integer
Statement-II: $(p+q)$ is an odd integer

Which one of the following is correct in respect of the above Question and the Statements?

A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Let $XYZ$ be a 3-digit number and the difference between $XYZ$ and $ZYX$ is equal to $PQR$. Is $(P+R)$ equal to $Q$?
Statement-I: $P=3$
Statement-II: $R=6$

Which one of the following is correct in respect of the above Question and the Statements?