Quantitative Aptitude · Mathematics

Numbers and Divisibility

215 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

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 $(x-1)$

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

If $(x^{100} + 2x^{99} + K)$ is exactly divisible by $(x + 1)$, find the value of 'K'

  1. $1$
  2. $2$
  3. $-2$
  4. $-3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x^{100}+2x^{99}+k$ is exactly divisible by $(x+1)$

$\therefore x=-1$ is the root of $x^{100}+2x^{99}+k$
$\Rightarrow (-1)^{100}+2(-1)^{99}+k=0$ 
$\Rightarrow 1-2+k=0$ 
$\Rightarrow \boxed{k=1}$

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

The sum of the digits of a 3 digit number is subtracted from the number. The resulting number is always.

  1. Divisible by 6

  2. Not divisible by 6

  3. Divisible by 9

  4. Not divisible by 9
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the no. be $xyz$ sum of digit is $(x+y+z)$ 

as $xyz=100x+10y+z$
then $xyz-(x+y+z)=99x+9y$  
$\therefore $ $\boxed{Always\, divisible\, by\, 9}$

Multiple choice multiplication and division vedic methods of multiplication vedic mathematics history of mathematics maths

The sum of all two digit numbers divisible by $5$ is:

  1. $1035$
  2. $1245$
  3. $1230$
  4. $945$

  5. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Required numbers are $10,15,20,25,......96$
This is an A.P. in which $a=10,d=5$ and $l=95$
${t} _{n}=95$ $\Rightarrow$ $a+(n-1)d=95$
$\Rightarrow$ $10+(n-1)\times 5=95$
$\Rightarrow$ $(n-1)\times 5=85$
$\Rightarrow$ $(n-1)=17$
$\Rightarrow$ $n=18$
$\therefore$ Required Sum $=\cfrac { n }{ 2 } (a+l)=\cfrac { 18 }{ 2 } \times (10+95)=(95\times 105)=945$

Multiple choice multiplication and division vedic methods of multiplication vedic mathematics history of mathematics maths

If $31z5$ is a multiple of $9$, where $z$ is a digit, what is the value of $z$?

  1. 0

  2. 4

  3. 7

  4. 9
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation

Given that $31z5$ is a multiple of $9$. 

According to the divisibility rule of $9$, the sum of all the digits should be a multiple of $9$. 
Therefore, 
$3 + 1 + z + 5 = 9\ OR\ 18$
$\Rightarrow z = 9 - 9 = 0$
$\Rightarrow z = 18 - 9 = 9$

Multiple choice maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

How many natural numbers are there between $23$ and $100$ which are exactly divisible by $24$?

  1. $8$
  2. $11$
  3. $12$
  4. $13$

  5. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Required numbers are $24,30,36,40,.....96$
This is an A.P. in which $a=24,d=6,l=96$
Let the number of terms in it be $n$.
Then ${t} _{n}=96$ $\Rightarrow$ $a+(n-1)d=96$
$\Rightarrow$ $24+(n-1)\times 6=96$
$\Rightarrow$ $(n-1)=12$
$\Rightarrow$ $n=13$
Required number of numbers $=13$

Multiple choice

What is the smallest positive integer that is divisible by 2, 3, and 5?

  1. 15

  2. 30

  3. 45

  4. 60
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The smallest positive integer that is divisible by 2, 3, and 5 is the least common multiple (LCM) of 2, 3, and 5. The prime factorization of 2 is (2), the prime factorization of 3 is (3), and the prime factorization of 5 is (5). Therefore, the LCM of 2, 3, and 5 is (2 \cdot 3 \cdot 5 = 30). Therefore, the smallest positive integer that is divisible by 2, 3, and 5 is 30.

Multiple choice

How many positive integers less than 100 are divisible by 3 or 5?

  1. 31

  2. 33

  3. 35

  4. 37
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

There are 33 positive integers less than 100 that are divisible by 3 or 5.

Multiple choice

What is the greatest common divisor (GCD) of 12 and 18?

  1. 2

  2. 3

  3. 6

  4. 9
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder. We can find the GCD of 12 and 18 by using the Euclidean algorithm: 18 = 12 * 1 + 6, 12 = 6 * 2 + 0. Therefore, the GCD of 12 and 18 is 6.

Multiple choice

What is the smallest positive integer that is divisible by 2, 3, and 5?

  1. 30

  2. 45

  3. 60

  4. 75
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The smallest positive integer that is divisible by 2, 3, and 5 is 30.

Multiple choice

Find the greatest common divisor (GCD) of 12 and 18.

  1. 2

  2. 3

  3. 6

  4. 9
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To find the greatest common divisor (GCD) of 12 and 18, we can use the Euclidean algorithm. We divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then we divide 12 by 6 to get a quotient of 2 and a remainder of 0. Therefore, the GCD of 12 and 18 is 6.