Quantitative Aptitude · Mathematics

Numbers and Divisibility

209 Questions

Master the number system by solving these quantitative aptitude questions on divisibility rules and properties. The exercises cover finding the greatest common divisor and identifying prime factors. This topic is essential for clearing the preliminary stages of SSC, banking, and various state exams.

Divisibility rulesGreatest common divisorPolynomial divisionNatural numbersFactorizationPrime numbers

Numbers and Divisibility Questions

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

The number of positive n in the range $12\le n\le 40$ such that the product (n -1) (n -2).... 3.2.1 is not divisible by n is :

  1. 5

  2. 7

  3. 13

  4. 14
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The product (n-1)! is divisible by n if n is composite and n > 4 (Wilson's Theorem related logic). For n in [12, 40], (n-1)! is not divisible by n only if n is prime. The primes between 12 and 40 are 13, 17, 19, 23, 29, 31, 37. There are 7 such numbers.

Multiple choice

Find the least number exactly divisible by 12, 15,20,27

  1. 520

  2. 530

  3. 540

  4. 510
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The least number divisible by 12, 15, 20, and 27 is their Least Common Multiple (LCM). The prime factorizations are 12=2^2*3, 15=3*5, 20=2^2*5, and 27=3^3; the LCM is 2^2*3^3*5 = 4*27*5 = 540.

Multiple choice

Which of the following is divisible by 3?

  1. 15287

  2. 15267

  3. 15286

  4. 152638
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A number is divisible by 3 if the sum of its digits is divisible by 3. For 15267, sum = 1+5+2+6+7 = 21, which is divisible by 3.

Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

What must be subtracted from or added to $8x^4+14x^3-2x^2+8x-12$ so that it may be exactly divisible by $4x^2+3x-2$?

  1. $15x-14$
  2. $3x-14$
  3. $-15x+14$
  4. $-3x+14$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$4x^2+3x-2)\overline {8x^4+14x^3-2x^2+8x-12}$ ( $2x^2+2x-1$
                            $\underline {\underset {-}{8}x^4\underset {-}{+}6x^3\underset {+}{-}4x^2}$
                            $8x^3+2x^2+8x-12$
                            $\underline {\underset {-}{8}x^3\underset {-}{+}6x^2\underset {+}{-}4x}$
                            $-4x^2+12x-12$
                            $\underline {\underset {+}{-}4x^2\underset {+}{-}3x\underset {-}{+}2}$
                                        $15x-14$

$\therefore$ The expression that must be subtracted is $15x-14$
and the expression that must be added is $-(15x-14)=-15x+14$

Multiple choice maths multiplication and division of algebraic expressions linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

$32x^{10}-33x^{5}+1$ is divisible by 

  1. $x-1$
  2. $x-2$
  3. $x-3$
  4. $x-4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$32x^{10}- 33 x^5 +1=0$
Let $m = x^5$
$\Rightarrow 32 m^2 - 33 m + 1 =0$
$\Rightarrow (32 m -1)(m-1)=0$
$\Rightarrow (32x^5-1)(x^5-1)=0$
$\therefore 32x^{10}- 33 x^5 +1=(32x^5-1)(x^5-1)$

$x^n-y^n$ is always divisible by $(x-y)$

$\therefore \,32x^{10}- 33 x^5 +1$ is divisible by $(x-1)$