# Number Theory and Number System Questions and Answers ### Problems on Number System:

Ques. A number when successively divided by 2, 5, 9 leaves 2, 3 and 4 as remainders respectively. If the number be divided by 9, 5 and 2 respectively, the respective remainders will be :
(a) 4, 3, 2
(b) 3, 4, 2
(c) 2, 4, 3
(d) 4, 2, 3

Ans. (a)

Ques. The largest natural number which exactly divides the product of any four consecutive natural numbers, is :
(a) 6
(b) 24
(c) 120
(d) 12

Ans. (c)

Ques. If 3th power of 9 is subtracted from 6th power of 2, then the digit in 10s place is
(a) 6
(b) 8
(c) 9
(d) 7

Ans. (c)

Ques. Consider the decimal 0.111010100010 …, when the digit in the “nth” decimal place is 1 if either n = 1 or n is a prime then the decimal
(a) terminates
(b) has only finite number of zeros
(c) recurs and denote a rational number
(d) neither terminates nor repeats.

Ans. (d)

Ques. A number when divided by 136 leaves 36 as remainder. If the same number is divided by 17, what will be the remainder:
(a) 2
(b) 3
(c) 5
(d) 15

Ans. (a)

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Ques. The traffic lights at 3 different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m. then the next change will appear after
(a) 7 m 14 sec
(b) 7 m 16 sec
(c) 7m 12 sec
(d) 7 m 18 sec

Ans. (c)

Ques. The largest number, which on dividing by 27, 77, and 202 leaves equal remainder in each case, is :
(a) 7
(b) 8
(c) 25
(d) 23

Ans. (c)

Ques. If the number 42573* is completely divisible by 72, then which number should replace the asterick(*)?
(a) 9
(b) 5
(c) 7
(d) 6

Ans. (d)

Ques. The representation of 2.317317317………….in the form of  p/q  is x and the representation of 18.484848…….. in form of a/b is y . Then x/y is
(a) 40626/5093
(b) 50926/6233
(c) 72341/52317
(d) None

Ans. (d)

Ques. If a number, of two digits, is 2 times the sum of its digits, the number formed by interchange the digits is equal to sum of the digits multiplied by :
(a) 1
(b) 8
(c) 9
(d) 4

Ans. (c)

Ques. The value of x, when 2x+4. 3x+1 = 288
(a) 1
(b) –1
(c) 0
(d) None of these

Ans. (a)

Ques. The product or the sum of two irrational numbers is always
(a) irrational
(b) rational
(c) neither rational nor irrational
(d) Either rational or irrational

Ans. (d)

Ques. If (874516)145689 is multiplied out, the units’ digit in the product :
(a) 4
(b) 3
(c) 6
(d) 1

Ans. (c)

Ques. The number of digits to the right of the decimal point in the product of 95.75 and 0.0255 is
(a) 5
(b) 6
(c) 7
(d) None of these

Ans. (b)

Ques. If (21567)125 is multiplied out, the units’ digit in the product is :
(a) 1
(b) 7
(c) 5
(d) 3

Ans. (b)

Related:

Ques. There are four composite numbers written in ascending order of magnitude. If the product of the first three is 192 and the last three is 432, then the first number is:
(a) 4
(b) 6
(c) 2
(d) 8

Ans. (a)

Ques. M and N are only two even numbers with M > N. The largest even integer which divides M2  -N2 is :
(a) 2
(b) 4
(c) 8
(d) 10

Ans. (d)

Ques. The least multiple of 5 which leaves a remainder 5 when divided by 6 or 10 or 15 is :
(a) 152
(b) 153
(c) 162
(d) 155

Ans. (d)

Ques. What is the least number which when divided by 16, 20, 32, 42, and 50 leaves the remainder 12, 16, 28,38, and 46 respectively?
(a) 11396
(b) 10396
(c) 10256
(d) 12546

Ans. (b)

Related:

Ques. For any natural number n(52n+2 – 24 n – 25) is divisible by :
(a) 25
(b) 50
(c) 510
(d) 576

Ans. (d)

Ques. 9876543210 is divisible by :
(a) 5, 9 and 11
(b) 5, 9 but not by 11
(c) 9 and 11 but not by 5
(d) 11 and 5 but not by 9

Ans. (b)

Ques. If n is odd, n(n2-1) is always divisible by :
(a) 18
(b) 20
(c) 24
(d) 32

Ans. (c)

Ques. If n is even number, then n(n2 + 10) is always divisible by:
(a) 10
(b) 4
(c) 6
(d) 28

Ans. (b)

Ques. If a number consists of units digit as a and the ten’s digit as b, such a number is:
(a) b + a
(b) bx
(c) 10b + x
(d) 10a + b

Ans. (b)

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Ques. If a number consists of units digit as 6 and the ten’s digit as 5, such a number is:
(a) 65
(b) 56
(c) 605
(d) 506

Ans. (b)

Ques. If n is even, n(n2-1) is always divisible by :
(a) 6
(b) 8
(c) 15
(d) 17

Ans. (a)

Ques. If a four digit number 2AB5 ( A and B  digits ) is divisible by 9, then the number AB is :
(a) 38
(b) 24
(c) 44
(d) 13

Ans. (a)

Ques. If the six digit number 4A52B6 is divisible by 2 as well as 3, then the number AB is :
(a) 22
(b) 21
(c) 20
(d) 12

Ans. (a)

Ques. If a number is divisible by 2 and 3 both, then it is necessarily divisible by :
(a) 6
(b) 8
(c) 9
(d) 12

Ans. (a)

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Ques. The smallest whole number by which 79633125 must be multiplied for the product to be a perfect square is :
(a) 11
(b) 5
(c) 13
(d) 3

Ans. (c)

Ques. The next number in the sequence 1/22,4/25, 9/28, 16/31, 25/34, is
(a) 28/35
(b) 30/37
(c) 36/37
(d) 36/35

Ans. (c)

Ques. The next number in the sequence 4/41, 5/51, 6/61, 7/71, is
(a) 8/81
(b) 9/81
(c) 8/75
(d) 8/78

Ans. (a)

Ques. The number of terms in the sequence 5, 9, 13, …………….189 is
(a) 45
(b) 46
(c) 48
(d) 47

Ans. (d)

Ques. The number of terms in the sequence 2, 9, 16,23 ……………. 338 is
(a) 48
(b) 46
(c) 49
(d) 45

Ans. (c)

Ques. What is the least number which when divided by 26, 34, 47, 58, and 112 leaves the remainder 16, 24, 37,48, and 102 respectively?
(a) 53282246
(b) 53582246
(c) 55326485
(d) 58565452

Ans. (a)

Ques. The least multiple of 9 which leaves a remainder 7 when divided by 9 or 13 or 15 is :
(a) 4102
(b) 4212
(c) 4302
(d) 4402

Ans. (a)

Ques. The tenth term of 8, 15, 22, 29,………..
(a) 81
(b) 71
(c) 51
(d) 91

Ans. (b)

Ques. The number 2.5252525………. as a fraction, when reduced to lowest terms, the sum of the numerator and denominator is
(a) 7
(b) 29
(c) 141
(d) 349

Ans. (d)

Ques. The seventh term of 112, 140, 168, …………..
(a) 270
(b) 250
(c) 280
(d) 300

Ans. (c)

Ques. P and Q are only two odd numbers with P > Q. The largest even integer which divides P2  -Q2 is :
(a) 2
(b) 8
(c) 4
(d) 6

Ans. (b)

Ques. If the square of a number of two digits is subtracted from the square of the number formed by interchanging the digits, the largest number by which the result is always divisible is :
(a) 9
(b) 99
(c) 11
(d) 15

Ans. (b)

Ques. The largest natural number by which the product of three consecutive odd numbers is always divisible is :
(a) 10
(b) 15
(c) 25
(d) 35

Ans. (b)

Ques. The smallest number which must be added to 803642 in order to obtain a multiple of 9 is
(a) 3
(b) 4
(c) 5
(d) 6

Ans. (b)

Ques. If (2158) 258 is multiplied out, the units’ digit in the product is:
(a) 3
(b) 5
(c) 6
(d) 4

Ans. (d)

Ques. If 5x–3 = 1, then the value of x is
(a) 2
(b) 3
(c) 1
(d) 1

Ans. (b)

Ques. Find the total number of prime numbers lying between 120 and 140.
(a) 3
(b) 4
(c) 5
(d) 6

Ans. (b)

Ques. If a and b are whole numbers such that ab = 169. The value of (a-3)b-1 is :
(a) 1
(b) 169
(c) 13
(d) 10

Ans. (d)

Ques. The least number which leaves remainders 2, 3, 4, 5 and 6 on dividing by 3, 4, 5, 6 and 7 respectively, is:
(a) 412
(b) 419
(c) 421
(d) 426

Ans. (b)

Ques. The total marks secured by Amit, Bhuvnesh and Chander are 540. Amit’s marks were three times that of Bhuvnesh’s and Bhuvnesh ‘s marks were twice that of Chander Amit’s marks are how much more than those of Chander ?
(a) 154
(b) 300
(c) 265
(d) 286

Ans. (b)

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Ques. In an exam, a student was asked to find 5/7 of a number. By mistake, she found 7/5 of it. Her answer was 96 more than the correct answer. The given number is
(a) 196
(b) 128
(c) 156
(d) 140

Ans. (d)

Ques. If x2 = 31423 , then x is
(a) an irrational number
(b) a complex number
(c) a rational number
(d) None

Ans. (a)

Ques. Two candies and a cookie cost Rs. 18 and two cookies and a candy cost Rs. 12. Find the cost of a candy.
(a) 8
(b) 2
(c) 16
(d) 6

Ans. (a)

Ques. The number of students in a school are 1125 more than 1/4th of it. Find the total strength.
(a) 1750
(b) 1500
(c) 2000
(d) 1225

Ans. (b)

Ques. Three Burger and a sandwiches cost 70 and 2 sandwiches and a Burger cost 40. How much will a Burger and a sandwiches cost?
(a) 10
(b) 20
(c) 30
(d) 40

Ans. (c)

Ques. The sum of 11 results is 55. The sum of the first 6 results is 30 and that of the last 6 is 48. Find the sixth result.
(a) 20
(b) 24
(c) 13
(d) 23

Ans. (d)

Ques. One-third of the total marks are required to pass an exam. A candidate who gets 178 marks, fails by 22 marks. The total marks in the exam are
(a) 550
(b) 600
(c) 535
(d) 660

Ans. (b)

Ques. At an election, a candidate who gets 3/4th of the total votes, is elected by a majority of 2000 votes. The total number of votes polled and the number of votes secured by the candidate who was elected, are respectively
(a) 4000, 3000
(b) 8000, 6000
(c) 4500, 2500
(d) 5000, 3000

Ans. (a)

Ques. Each student in a class contributed as many paise as the number of students in the class. The teacher contributed \$13 to make the total collection of \$49. How many students are there in the class?
(a) 70
(b) 50
(c) 60
(d) 65

Ans. (c)

Ques. A box contains 44 more than 1/5th of total number of oranges. How many oranges are there in the box?
(a) 66
(b) 55
(c) 65
(d) 33

Ans. (b)

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Ques. A water tank having 1300 litres of capacity was filled by adding 75 litres of water and as many buckets of water as each bucket had a capacity. What was the capacity of each bucket in litres ?
(a) 32
(b) 35
(c) 48
(d) 45

Ans. (b)

Ques. The sum of four consecutive numbers is 90. Find the difference between five times the third number and 4 times the fourth number.
(a) 21
(b) 29
(c) 24
(d) 19

Ans. (d)

Ques. The smallest number which must be subtracted from 1300 to make it a perfect square is
(a) 2
(b) 3
(c) 4
(d) 6

Ans. (c)

Ques. In an exam, a student was asked to find 4/5th  of a number. By mistake, he found 5/4th of it and his answer was180 more than the correct answer. Find the given number.
(a) 810
(b) 890
(c) 400
(d) 500

Ans. (c)

Ques. In a school library,1/2 of the books are story books, 2/3rd of the remaining books are reference books and the remaining 250 books are encyclopaedias. Find the total number of books in the library.
(a) 1,200
(b) 1,500
(c) 2,000
(d) 750

Ans. (b)

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Ques. 4/5th of a number exceeds its 2/3rd of 9/10th by 120. Find the number.
(a) 600
(b) 140
(c) 800
(d) 660

Ans. (a)

Ques. 5/8 th of 120% of a number is 235 more than 245. Find that number.
(a) 523
(b) 684
(c) 576
(d) 640

Ans. (d)

Ques. 24 is divided into two parts such that 6 times the first part exceeds 4 times the second part by 4. Find the first part.
(a) 14
(b) 16
(c) 10
(d) 12

Ans. (c)

Ques. The amount that Satheesh had was \$675 more than 3/5th of 1/9th of \$450. Find the amount.
(a) 700
(b) 705
(c) 625
(d) 605

Ans. (b)

Ques. 340 students are arranged in a parade such that the number of students in each column is the same as the number of columns. Find how many students are excluded from the group to make such an arrangement possible.
(a) 16
(b) 21
(c) 14
(d) 24

Ans. (a)

Ques. A sum is divided between Rajiv and Raghav such that 5/6th of the amount of Rajiv is equal to 4/5th the amount of Raghav. If Rajiv gets \$240, find the sum.
(a) 500
(b) 250
(c) 490
(d) 400

Ans. (c)

Ques. The total age of 9 girls is 128. The total age of the first 4 girls is 62 and that of the next 4 girls is 48. Find the age of the 9th girls.
(a) 16
(b) 18
(c) 12
(d) 17

Ans. (b)

Ques. A sum is divided between Vijay and Aman such that3/5th of the amount of Vijay is equal to5/7th of the amount of Aman. If Vijay gets \$750, how much does Aman get?
(a) 450
(b) 600
(c) 630
(d) 730

Ans. (c)

Ques. Sum of two numbers is 1/3rd of 1/5th of 195 and their product is 1/6th  of 1/4th of 960. Find 1/3rd of the difference between them.
(a) 1
(b) 9
(c) 3
(d) 27

Ans. (a)

Ques. In a country, 5/9th of the population are adults.1/2 of the adults are male, 3/5th of the adult females are illiterate. If 800 adult females are illiterate, then the population of the village is
(a) 4,000
(b) 4,800
(c) 9,000
(d) 5,600

Ans. (b)

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Ques. When 6 is added to a number and the sum is multiplied by 8, the result is same as when 26 is multiplied by 10 and 12 is added to the product. The number is
(a) 28
(b) 23
(c) 26
(d) 25

Ans. (a)

Ques. If x2 = 84681, then x is
(a) an irrational number
(b) a rational number
(c) a complex number
(d) None of these

Ans. (b)

Ques. The sum of three consecutive even integers is 132. Find the difference between 3 times the least and 1/2 of the greatest.
(a) 44
(b) 89
(c) 144
(d) 103

Ans. (d)

Ques. 7/8 th of a number exceeds its 7/16 th by 84. Find 3/4th of the number.
(a) 144
(b) 192
(c) 138
(d) 204

Ans. (a)

Ques. The conjugate of √7 + 4 is
(a) √7 – 4
(b) -√7 – 4
(c) -√7 + 4
(d) All the above

Ans. (c)

Ques. Which of the following is divisible by 9?
(a) 1278
(b) 1754
(c) 1779
(d) 1770

Ans. (a)

Ques. The least perfect square, which is divisible by each of 21, 36 and 66 is
(a) 214344
(b) 214434
(c) 213444
(d) 231444

Ans. (c)

Ques. The number of rationalizing factors of an irrational number is
(a) One
(b) Two
(c) Three
(d) Infinite

Ans. (d)

#### Yash Bansal

Yash is co-founder of Examsegg. Yash loves to solve questions on reasoning and mathematics. He taught himself how to blend different techniques to get accurate solution in lesser time.