# Problems on Numbers

Candidates can solve the given aptitude test using Problems on Numbers Shortcut Tricks. Problems on Numbers Questions with Answers are provided here, so that individuals can prepare well for this topic and score good marks in the examination. These Problems on Numbers with solutions will helps you at the time of practice.

Candidates can match out their selected option with the answers that are provided at the end of this page. Explanation of important and selected Problems on Numbers Aptitude Test Questions is also provided here, so that individuals can go through these questions. Students can also download Problems on Numbers Questions PDF from below section of this page.

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## Problems on Numbers

Problems on Numbers Concept

1. Natural Numbers:

Counting numbers 1, 2, 3, 4, 5, .. are called natural numbers.

1. Whole Numbers:

All counting numbers together with zero form the set of whole numbers. Thus,

1. 0 is the only whole number which is not a natural number.
2. Every natural number is a whole number.

Problems on Numbers Basic Formulas

1) (a – b)2 = (a2 + b2 – 2ab)

2) (a + b)2 = (a2 + b2 + 2ab)

3) (a + b) (a – b) = (a2 – b2 )

4) (a3 + b3) = (a + b) (a2 – ab + b2)

5) (a3 – b3) = (a – b) (a2 – ab + b2)

6) (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca)

7) (a3 + b3 + c3 – 3abc) = (a + b + c) (a2 + b2 + c2 – ab – bc – ac)

8) When a + b + c = 0, then a3 + b3 + c3 = 3abc

9) (a + b)n = an + (nC1)an-1b + (nC2)an-2b2 + … + (nCn-1)abn-1 + bn

Problems on Numbers Shortcut Tricks

1) 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2

2) (12 + 22 + 32 + ….. + n2) = n ( n + 1 ) (2n + 1) / 6

3) (13 + 23 + 33 + ….. + n3) = (n(n + 1)/ 2)2

4) Sum of first n odd numbers = n2

5) Sum of first n even numbers = n (n + 1)

6) A number is divisible by 2, if its unit’s digit is any of 0, 2, 4, 6, 8.

7) A number is divisible by 3, if the sum of its digits is divisible by 3.

8) A number is divisible by 4, if the number formed by the last two digits is divisible by 4.

9) A number is divisible by 5, if its unit’s digit is either 0 or 5.

10) A number is divisible by 6, if it is divisible by both 2 and 3.

11) A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.

12) A number is divisible by 9, if the sum of its digits is divisible by 9.

13) A number is divisible by 10, if it ends with 0.

14) A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.

15) A number is divisible by 12, if it is divisible by both 4 and 3.

16) A number is divisible by 14, if it is divisible by 2 as well as 7.

17) Two numbers are said to be co-primes if their H.C.F. is 1. To find if a number, say y is divisible by x, find m and n such that m * n = x and m and n are co-prime numbers. If y is divisible by both m and n then it is divisible by x.

Problems on Numbers Questions With Answers

Ques1: What is the place value of 6 in 64?

A) 6

B) 60

C) 64

D) 10

 Explanation:We know that,Each digit has a fixed position called its place.Each digit has a value depending on its place called the place value of the digit.The face value of a digit for any place in the given number is the value of the digit itselfPlace value of a digit = (face value of the digit) × (value of the place).Hence, the place value of 6 in 64 = 6 x 10 = 60.

Ques2: In a two-digit number, the digit in the unit’s place is more than twice the digit in ten’s place by 1. If the digits in the unit’s place and the ten’s place are interchanged, difference between the newly formed number and the oiginal number is less than the original number by 1. What is the original number ?

A) 35

B) 36

C) 37

D) 39

 Explanation:Let the ten’s digit be x. Then, unit’s digit = 2x + 1.[10x + (2x + 1)] – [{10 (2x + 1) + x} – {10x + (2x + 1)}] = 1<=> (12x + 1) – (9x + 9) = 1 <=> 3x = 9, x =  3.So, ten’s digit = 3 and unit’s digit = 7. Hence, original number = 37.

Ques3: The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is :

A) 100

B) 200

C) 300

D) 400

 Explanation:Let the numbers be x and y.Then, xy = 9375 and  x/y = 15.xyxy=9375/15=>y2 =625=> y = 25=> x = 15y = 15 x 25 = 375.Sum of the numbers = 375 + 25 = 400.

Ques4: Three numbers are in the ratio 4 : 5 : 6 and their average is 25. The largest number is :

A) 30

B) 40

C) 50

D) 60

 Explanation:Let the numbers be 4x, 5x and 6x, Then, (4x + 5x + 6x ) / 3 = 25=> 5x = 25=> x = 5.Largest number  6x = 30.

Ques5: In a two-digit number, if it is known that its unit’s digit exceeds its ten’s digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is :

A) 12

B) 24

C) 36

D) 48

 Explanation:Let the ten’s digit be x.Then, unit’s digit = x + 2.Number = 10x + (x + 2) = 11x + 2Sum of digits = x + (x + 2) = 2x + 2.(11x + 2) (2x + 2) = 144=> 22x^2+26x-140=0=> (x – 2)(11x + 35) = 0=> x = 2Hence, Required Number = 11x + 2 = 24

Ques6: Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is :

A) 12

B) 137

C) 14

D) 15

 Explanation:Let the three integers be x, x + 2 and x + 4. Then, 3x = 2 (x + 4) + 3  <=> x = 11.Third integer = x + 4 = 15.

Ques7: The denominator of a fraction is 3 more than the numerator. If the numerator as well as the denominator is increased by 4, the fraction becomes 4/5. What was the original fraction ?

A) 7/11

B) 8/11

C) 9/11

D) 10/11

 Explanation:Let the numerator be x Then, denominator =  x + 3.Now. (x + 4)/(x + 3) +4 = 4/5  <=> 5 (x + 4) =  4(x + 7)=>  x = 8.The fraction is  8/11.

Ques8: 54 is to be divided into two parts such that the sum of 10 times the first and 22 times the second is 780. The bigger part is :

A) 24

B) 34

C) 44

D) 54

 Explanation:Let the two parts be (54 – x) and x.Then, 10 (54 – x) + 22x = 780=> 12x = 240=>  x = 20.Bigger part = (54 – x)  = 34.

Ques9: (3x + 2) (2x – 5) = ax² + kx + n . What is the value of a – n + k ?

A) 5

B) 8

C) 9

D) 10

 Explanation:This is a quadratic equation. To find n we multiply the last terms in each bracket together (2 times -5) = -10To find ‘a’ we multiply the first terms in each bracket together (3x times 2x) = 6×2, So a = 6.Thus a – n = 16To find k we multiply the first term in the first bracket by the second term in the second bracket and add the result to the product of the second term in the first bracket and the first term in the second. (-15x + 4x) = -11x. So k = -1116 – 11 = 5

Ques10: There are two numbers such that the sum of twice the first and thrice the second is 39, while the sum of thrice the first, and twice the second is 36. The larger of the two is :

A) 3

B) 6

C) 9

D) 12

 Explanation:Let the numbers be x and y. Then, 2x + 3y = 39…(i)and 3x + 2y = 30 …(ii)On solving (i) and (ii), we get :  x = 6 and y = 9.larger number = 9

Ques11: The sum of the squares of two numbers is 3341 and the diference of their squares is 891. The numbers are :

A) 35 and 46

B) 35 and 50

C) 40 and 55

D) 45 and 60

 Explanation:Let the numbers be x and y. Then,x^2+y^2=3341……(1)x^2-y^2=891……(2)Adding (i) and (ii), we get : 2x^2=4232 or x^2= 2116 or x =46Subtracting (ii) from (i), we get : 2y^2= 2450 or y^2 = 1225 or y = 35So, the numbers are 35 and 46

Ques12: The product of three consecutive even numbers when divided by 8 is 720. The product of their square roots is :

A) 12 sqrt(10)

B) 24 sqrt(10)

C) 120

D) none of these

 Explanation:Let the numbers be x, x + 2 and X + 4.Then, x(x + 2)(x +4) / 8 = 720 => x(x + 2)(x +4) = 5760.√x  *  √(x + 2) * √(x +4) = √x(x +2)(x + 4 ) = √5760  =  24√10.

Ques13: The difference between a two-digit number and the number obtained by interchanging the two digits is 63. Which is the smaller of the two numbers ?

A) 12

B) 15

C) 17

D) none of these answers can be determined

 Explanation:Let the ten’s digit be x and unit’s digit be y.Then, (10x + y) – (10y + x) = 63=>  9 (x – y) = 63=> x – y = 7.Thus, none of the numbers can be determined..

Ques14: There are four prime numbers written in ascending order of magnitude. The product of the first three is 385 and of the last three is 1001. Find the fourth number ?

A) 17

B) 19

C) 13

D) 11

 Explanation:(385/1001) = 5/13First Number is 5=> Fourth number is 13

Ques15: The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. What is the two-digit number?

A) 69

B) 78

C) 96

D) Cannot be determined

E) None of these

 Explanation:Let the ten’s digit be x and unit’s digit be y.Then, x + y = 15 and x – y = 3   or   y – x = 3.Solving x + y = 15   and   x – y = 3, we get: x = 9, y = 6.Solving x + y = 15   and   y – x = 3, we get: x = 6, y = 9.So, the number is either 96 or 69.Hence, the number cannot be determined.

Ques16: If one-third of one-fourth of a number is 15, then three-tenth of that number is:

A) 35

B) 36

C) 45

D) 54

 Explanation:Let the number be x.Then,  1/3 of 1/4 of x = 15<=> x = 15 x 3 x 4 = 180.So, required number = 3/10*180 = 54.

Ques17. Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:

A) 9

B) 11

C) 13

D) 15

 Explanation:Let the three integers be x, x + 2 and x + 4.Then, 3x = 2(x + 4) + 3<=>x = 11.Third integer = x + 4 = 15.

Ques18. The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?

A) 3

B) 4

C) 9

D) Cannot be determined

E) None of these

 Explanation:Let the ten’s digit be x and unit’s digit be y.Then, (10x + y) – (10y + x) = 369(x – y) = 36x – y = 4.

Ques19. The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2 ?

A) 4

B) 8

C) 16

D) None of these

 Explanation:Since the number is greater than the number obtained on reversing the digits, so the ten’s digit is greater than the unit’s digit.Let ten’s and unit’s digits be 2x and x respectively.Then, (10 x 2x + x) – (10x + 2x) = 369x = 36x = 4.Required difference = (2x + x) – (2x – x) = 2x = 8.

Ques20. A two-digit number is such that the product of the digits is 8. When 18 is added to the number, then the digits are reversed. The number is:

A) 18

B) 24

C) 42

D) 81

 Explanation:Let the ten’s and unit digit be x and 8/x respectively.Then, 10x + 8/x + 18 = 10×8/x + x10x2 + 8 + 18x = 80 + x29x2 + 18x – 72 = 0x2 + 2x – 8 = 0(x + 4)(x – 2) = 0x = 2