The prime factorization of $15$ is

Important Questions on Factors and Rational Numbers

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

Find the smallest positive integer $n\ge 10$ such that $n+6$ is a prime and $9n+7$ is a perfect square.

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

A two-digit number $\overline{ab}$ is called almost prime if one obtains a two-digit prime number by changing at most one of its digits $a \& b$. (For example, 18 is an almost prime number because 13 is a prime number). Then the number of almost prime two-digit numbers is

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

There are four prime numbers written in ascending order. The product of the first three is $385$ and that of the last three is $1001.$ The last number is :

HARD

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

The prime numbers $a, b$ and $c$ are such that $a+b^2=4c^2.$ Determine the sum of all possible values of $a+b+c.$

MEDIUM

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

For an integer $n$, let $S_n=\left\{n+1,n+2,\dots ., n+18\right\}$. Which of the following is true for all $n\ge 10$ ?

HARD

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p>\sqrt{\mathit{n}}$ and $p<20.$ What is the value of $N?$

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

A prime number $p$ is called special if there exist primes $p_1, p_2, p_3, p_4$ such that $p=p_1+p_2=p_3-p_4.$
The number of special primes is
 

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

Express $120$ as a product of its prime factors.

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

Which of the following number is an odd and prime number?

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

The sum of the squares of the digits of the largest two-digit prime number is:

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

If $a$ and $b$ are positive integers such that $a^2-b^2$ is a prime number, then $a^2-b^2$ is equal to:

HARD

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

The number of $6$-digit numbers of the form $ababab$ (in base $10$) each of which is a product of exactly $6$ distinct primes is

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

It is given that the number $43361$ can be written as a product of two distinct prime numbers $p_1, p_2.$ Further, assume that there are $42900$ numbers which are less than $43361$ and are co-prime to it. Then, $p_1+p_2$ is

MEDIUM

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

Consider the set $E=\{5,6,7,8,9\}.$ For any partition $\{A,B\}$ of $E,$ with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B.$ Let $N$ be the largest prime number among these numbers. Find the sum of the digits of $N.$

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

Express $120$ as a product of its prime factors.

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

The number of prime numbers which are less than $100$ is:

HARD

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

For $a,b\in \left\{0,1,\cdots ,9\right\}$, a two digit number $ab$ is defined for $10a+b$. A two digit number $ab$ is called a special prime if both $ab$ and $ba$ are primes. Then the total number of special primes is

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

Which of the following is not a pair of prime numbers ?

HARD

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

Suppose $a,b,c$ are positive integers such that $2^a+4^b+8^c=328.$ Then $\frac{a+2b+3c}{abc}$ is equal to-

EASY

Mathematics>Arithmetic>Factors and Rational Numbers>Prime Factors

If $a$ and $b$ are positive integers such that $\left(a^2-b^2\right)$ is a prime number, then