Mathematics · Quantitative Aptitude

Number Theory

438 Questions

Number theory explores the properties and relationships of integers, focusing heavily on prime and composite numbers. Key areas include identifying prime numbers, understanding co primes, and exploring mathematical conjectures like Goldbach's. These questions are a core component of quantitative aptitude sections in banking and government exams.

Prime number identificationCo prime propertiesGoldbach conjectureOdd and even numbersDivisibility rules

Number Theory Questions

Multiple choice

Give the prime factorization of 1260

  1. 2 x 32 x 52 x 7

  2. 22 x 32 x 5 x 7

  3. 23 x 3 x 5 x 7

  4. 22 x 3 x 52 x 7
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

[Explanation: Prime factorization of 1260 can be calculated using division method. 1260= 2 x 2 x 3 x 3 x 5 x 7= 22 x 32 x 5 x 7]

Multiple choice

x and y are integers. x has 3 prime factors (excludes x itself) and y has 4 prime factors (excludes y itself). If the greatest common factor of x and y is 21, how many prime factors does x*y have?

  1. 2

  2. 5

  3. 21

  4. 4
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

21 is the greatest common factor of x and y. So 3 and 7 are two common 2 prime factors of x and y. Hence, x still has one prime factor a and y still has two prime factors b and c. The prime factors of x*y should be a, b, c, 3 and 7, the total number is 5.

Multiple choice

x is a prime number that is less than 7, what is x? 1) 8^ (- x) < 2^(- 12) 2) x > 4

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

  4. EACH statement ALONE is sufficient.

Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

1) only, 8^ (- x) = 2^ (- 3x) < 2^ (- 12) so - 3x < - 12, x > 4. Because x should be less than 7 and x is a prime number, x must be 5, sufficient 2) only, x > 4 and x < 7, x is prime number, x must be 5, sufficient.

Multiple choice

If x is a positive integer, does it have more than 4 different prime factors? 1) x < 210 2) x > 210

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

  4. EACH statement ALONE is sufficient.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

assume y = 2*3*5*7 = 210, y has 4 prime factors and y is the least possible value which has 4 different prime factors. So if x has more than 4 different prime factors, x must be more than 210.  1) only, x < 210, x cannot have more than 4 prime factors, sufficient. 2) only, x > 210, x may have more than 4 different prime factors, but x may have only 4 prime different factors, insufficient.

Multiple choice

What is the smallest prime number?

  1. 2

  2. 144

  3. 1

  4. 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The smallest such number is 2.