Chapter 2: NUMBER THEORY


Introduction


Highest common factor :


If there are three numbers like 10, 20, 30 and there exists numbers that can divide them to give remainder 0 viz. 5, 10. Then the lowest of those numbers is the HCF viz. 5


GCD and HCF solving techniques


Least Common multiple:


If we take 4,6 as two numbers then both of them divide 12, 24, 36 etc fully giving remainder 0 so the lowest such number common to both is the LCM which in this case is 12.


Method 1:



Method 2:



Questions:

Q. What is the least number of cans needs for a vendor who has 21 L of Milk 'a' , 42 L of milk 'b' and 63L of Milk 'c'. If all cans should have same number of milk and he should need least number of cans.


Ans: Find HCF of 21, 42, 63 which is 21. So total cans are 6 i.e. 1 for milk 'a' , 2 for milk 'b' and 3 for milk 'c'.


Q. If the HCF and LCM of two numbers is 18, 180 and one of the number is 36. Then the second number is.


Ans. HCF * LCM = a * but HCF = 18, LCM = 180 and a= 36 so substituting we get b = 90.


CAT Problems


Q.The smallest number which when divided by 4, 6 or 7 leaves a remainder of 2, is


  1. 44
  2. 62
  3. 80
  4. 86

Ans.d


Q.A young girl counted in the following way on the fingers of her left hand. She started calling the thumb 1, the index finger 2, middle finger 3, ring finger 4, little finger 5, then reversed direction, calling the ring finger 6, middle finger 7, index finger 8, thumb 9, then back to the index finger for 10, middle finger for 11, and so on. She counted up to 1994. She ended on her


  1. thumb
  2. index finger
  3. middle finger
  4. ring finger

Ans.b


Q.A report consists of 20 sheets each of 55 lines and each such line consist of 65 characters. This report is retyped into sheets each of 65 lines such that each line consists of 70 characters. The percentage reduction in number of sheets is closest to


  1. 20
  2. 5
  3. 30
  4. 35

Ans.a


Q.Which is the least number that must be subtracted from 1856, so that the remainder when divided by 7, 12 and 16 is 4.


  1. 137
  2. 1361
  3. 140
  4. 172

Ans.d


Q. If a,b are consecutive even numbers their LCM is a*b/2;


Q. HCF of fractions = HCF of numerator / LCM of denominator;


Q. LCM of fractions = LCM of numerators / HCF of denominators;


Q. Number of zeroes at the end of the factorial = n!
            [ n/5 ] + [ n/5^2 ] + [ n/5^3 ] ....

Number theory practice - CAT


A series S1 of five positive integers is such that the third term is half the first term and the fifth term is 20 more than the first term. In series S2, the nth term defined as the difference between the (n+1)th term and the nth term of series S1, is an arithmetic progression with a common difference of 30.

Q.First term of S1 is


  1. 80

  2. 90

  3. 100

  4. 120


Ans.c


Q.Second term of S2 is


  1. 50

  2. 60

  3. 70

  4. none


Ans.d


Q.What is the difference between second and fourth terms of S1?


  1. 10

  2. 20

  3. 30

  4. 60


Ans.a


Q.What is the average value of the terms of series S1?


  1. 60

  2. 70

  3. 80

  4. average isnt an integer


Ans.c


Q.What is the sum of series S2?


  1. 10

  2. 20

  3. 30

  4. 40


Ans.b


Q.If n is any odd number greater than 1, then n(n2 – 1) is


  1. divisible by 96 always

  2. divisible by 48 always

  3. divisible by 24 always

  4. None of these


Ans.c


Q.If a number 774958A96B is to be divisible by 8 and 9, the respective values of A and B will be


  1. 7,8

  2. 8,0

  3. 5,8

  4. none


Ans.b


Q.Which of the following values of x do not satisfy the inequality (x2 – 3x + 2 > 0) at all?


  1. 1 ≤ x ≤ 2

  2. –1 ≥ x ≥ –2

  3. 0 ≤ x ≤ 2

  4. 0 ≥ x ≥ –2


Ans.a


Q. What is the digit in the unit’s place of 251?


  1. 2

  2. 8

  3. 1

  4. 4


Ans . B


Q. A number is formed by writing first 54 natural numbers in front of each other as 12345678910111213 ... Find the remainder when this number is divided by 8.


  1. 1

  2. 7

  3. 2

  4. 0


Ans . C


Q. How many five-digit numbers can be formed using the digits 2, 3, 8, 7, 5 exactly once such that the number is divisible by 125?


  1. 0

  2. 1

  3. 4

  4. 3


Ans . C


Q. You can collect as many rubies and emeralds as you can. Each ruby is worth Rs. 4 crore and each emerald is worth Rs. 5 crore. Each ruby weighs 0.3 kg. And each emerald weighs 0.4 kg. Your bag can carry at the most 12 kg. What should you collect to get the maximum wealth?


  1. 20 rubies and 15 emeralds

  2. 40 rubies

  3. 28 rubies and 9 emeralds

  4. None of these


Ans . B


Q. I have one-rupee coins, 50-paisa coins and 25-paisa coins. The number of coins are in the ratio 2.5 : 3 : 4. If the total amount with me is Rs. 210, find the number of one-rupee coins


  1. 90

  2. 85

  3. 100

  4. 105


Ans . D


Q. My son adores chocolates. He likes biscuits. But he hates apples. I told him that he can buy as many chocolates he wishes. But then he must have biscuits twice the number of chocolates and should have apples more than biscuits and chocolates together. Each chocolate cost Re. 1. The cost of apple is twice the chocolate and four biscuits are worth one apple. Then which of the following can be the amount that I spent on that evening on my son if number of chocolates, biscuits and apples brought were all integers?


  1. 34

  2. 33

  3. 8

  4. none


Ans . A


Q. A certain number, when divided by 899, leaves a remainder 63. Find the remainder when the same number is divided by 29.


  1. 5

  2. 4

  3. 1

  4. cant say


Ans . A


Q. A is the set of positive integers such that when divided by 2, 3, 4, 5, 6 leaves the remainders 1, 2, 3, 4, 5 respectively. How many integers between 0 and 100 belong to set A?


  1. 0

  2. 1

  3. 2

  4. none


Ans . B


Q. In a survey of political preferences, 78% of those asked were in favour of at least one of the proposals: I, II and III. 50% of those asked favoured proposal I, 30% favoured proposal II and 20% favoured proposal III. If 5% of those asked favoured all three of the proposals, what percentage of those asked favoured more than one of the three proposals?


  1. 10

  2. 12

  3. 17

  4. 22


Ans . C


Q. For two positive integers a and b define the function h(a,b) as the greatest common factor (G.C.F) of a, b. Let A be a set of n positive integers. G(A), the G.C.F of the elements of set A is computed by repeatedly using the function h. The minimum number of times h is required to be used to compute G is


  1. n/2

  2. n-1

  3. n

  4. none of these


Ans . B


Q. Let D be a recurring decimal of the form D = 0. a1 a2 a1 a2 a1 a2 ..., where digits a lie between 0 and 9. Further, at most one of them is zero. Which of the following numbers necessarily produces an integer, when multiplied by D?


  1. 18

  2. 108

  3. 198

  4. 288


Ans . C


Q. In the above table, for suitably chosen constants a, b and c, which one of the following best describes the relation between y and x?

arithmatic
  1. y = a + bx

  2. y = a + bx + cx2

  3. y = ea+ bx

  4. None of these


Ans . B


Q. if a1 = 1, an+1= 2 * an + 5 ; n=1.2... then a100 is


  1. (5 × 299 - 6)

  2. (5 × 299 + 6)

  3. (6 × 299 + 5)

  4. (6 × 299 - 5)


Ans . D


Q. What is the value of the following expression?

arithmatic
  1. 9/19

  2. 10/19

  3. 10/21

  4. 11/21


Ans . C


Q. Consider a sequence of seven consecutive integers. The average of the first five integers is n. The average of all the seven integers is


  1. n

  2. n + 1

  3. k × n, where k is a function of n

  4. n + 2/7


Ans . B


Q. If x > 2 and y > –1, then which of the following statements is necessarily true?


  1. xy > –2

  2. –x < 2y

  3. xy < –2

  4. –x > 2y


Ans . B


Q. Let S be the set of integers x such that
I. 100 ≤ x ≤ 200
II. x is odd and
III. x is divisible by 3 but not by 7.
How many elements does S contain?


  1. 16

  2. 12

  3. 11

  4. 13


Ans . D


Q. Let x, y and z be distinct integers, that are odd and positive. Which one of the following statements cannot be true?


  1. xyz2 is odd

  2. (x – y)2 z is even

  3. (x + y – z)2 (x + y) is even

  4. (x – y)(y + z)(x + y – z) is odd


Ans . D


Q. Let S be the set of prime numbers greater than or equal to 2 and less than 100. Multiply all elements of S. With how many consecutive zeros will the product end?


  1. 1

  2. 4

  3. 5

  4. 10


Ans . A


Q. Let N = 1421 × 1423 × 1425. What is the remainder when N is divided by 12?


  1. 0

  2. 9

  3. 3

  4. 6


Ans . C


Q. The integers 34041 and 32506, when divided by a three-digit integer n, leave the same remainder. What is the value of n?


  1. 289

  2. 367

  3. 453

  4. 307


Ans . D


Q. read and answer

quantitative aptitude
  1. n is even

  2. n is odd

  3. n is an odd multiple of 3

  4. n is prime


Ans . A


Q. Answer the questions based on the following information.

A, B and C are three numbers. Let
@ (A, B) = Average of A and B,
/ (A, B) = Product of A and B, and
× (A, B) = The result of dividing A by B.


Q. The sum of A and B is given by


  1. / (@ (A, B), 2)

  2. × (@ (A, B), 2)

  3. @ (/ A, B), 2)

  4. @ (× (A, B), 2)


Ans . A


Q. Average of A, B and C is given by


  1. @ (/ (@ (/ (B, A), 2), C), 3)

  2. × (@ (/ (@ (B, A), 3), C), 2)

  3. / (×(× (@ (B, A), 2), C), 3)

  4. / (× (@ (/ (@ (B, A) 2), C), 3), 2)


Ans . D


Q. Answer the questions based on the following information.

quantitative aptitude

Q. Which of the following expressions yields a positive value for every pair of non-zero real numbers (x, y)?


  1. f(x, y) – g(x, y)

  2. f(x, y) – (g(x, y))2

  3. g(x, y) – (f(x, y))2

  4. f(x, y) + g(x, y)


Ans . D


Q. Under which of the following conditions is f(x, y) necessarily greater than g(x, y)?


  1. Both x and y are less than –1

  2. Both x and y are positive

  3. Both x and y are negative

  4. y > x


Ans . A


Q. Convert the number 1982 from base 10 to base 12. The result is


  1. 1182

  2. 1912

  3. 1192

  4. 1292


Ans . C


Q. For all non-negative integers x and y, f(x, y) is defined as below.
f(0, y) = y + 1
f(x + 1, 0) = f(x, 1)
f(x + 1, y + 1) = f(x, f(x + 1, y))
Then what is the value of f(1, 2)?


  1. 2

  2. 4

  3. 3

  4. CANT SAY


Ans . B


Q. The set of all positive integers is the union of two disjoint subsets:
{f(1), f(2), ..., f(n), ...} and {g(1), g(2), ..., g(n), ...}, where
f(1) < f(2) < ... < f(n) ..., and g(1) < g(2) < ... < g(n) ..., and
g(n) = f(f(n)) + 1 for all n ≥ 1.
What is the value of g(1)?


  1. 0

  2. 2

  3. 1

  4. cant determine


Ans . B


Q. If the equation x3 – ax2 + bx – a = 0 has three real roots, then it must be the case that


  1. b = 1

  2. b ≠ 1

  3. a = 1

  4. a ≠ 1


Ans . B


Q. Let N = 553 + 173– 723. N is divisible by


  1. both 7 and 13

  2. both 3 and 13

  3. both 17 and 7

  4. both 3 and 17


Ans . D


Q. If x2 + y2 = 0.1 and |x – y| = 0.2, then |x| + |y| is equal to


  1. 0.3

  2. 0.4

  3. 0.2

  4. 0.6


Ans . B


Q. Let x, y and z be distinct integers. x and y are odd and positive, and z is even and positive. Which one of the following statements cannot be true?


  1. y(x – z)2 is even

  2. y2(x – z) is odd

  3. y(x – z) is odd

  4. z(x – y)2 is even


Ans . A


Q. If x > 5 and y < –1, then which of the following statements is true?


  1. (x + 4y) > 1

  2. x > –4y

  3. –4x < 5y

  4. None of these


Ans . D


Q. In a four-digit number, the sum of the first 2 digits is equal to that of the last 2 digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other 2 digits. What is the third digit of the number?


  1. 5

  2. 8

  3. 1

  4. 4


Ans . A


Q. Anita had to do a multiplication. Instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?


  1. 1050

  2. 540

  3. 1440

  4. 1590


Ans . D


Q. x and y are real numbers satisfying the conditions 2 < x < 3 and – 8 < y < –7. Which of the following expressions will have the least value?


  1. x2y

  2. xy2

  3. 5xy

  4. none


Ans . C


Q. m is the smallest positive integer such that for any integer n ≥ m , the quantity n3 -72+11n - 5 is positive. What is the value of m?


  1. 4

  2. 5

  3. 8

  4. none


Ans . D


Q. Three friends, returning from a movie, stopped to eat at a restaurant. After dinner, they paid their bill and noticed a bowl of mints at the front counter. Sita took one-third of the mints, but returned four because she had a momentary pang of guilt. Fatima then took one-fourth of what was left but returned three for similar reason. Eswari then took half of the remainder but threw two back into the bowl. The bowl had only 17 mints left when the raid was over. How many mints were originally in the bowl?


  1. 38

  2. 31

  3. 41

  4. none


Ans . D


Q. In a number system, the product of 44 and 11 is 3414. The number 3111 of this system, when converted to the decimal number system, becomes


  1. 406

  2. 1086

  3. 213

  4. 691


Ans . A


Q. All the page numbers from a book are added, beginning at page 1. However, one page number was added twice by mistake. The sum obtained was 1000. Which page number was added twice?


  1. 44

  2. 45

  3. 10

  4. 12


Ans . C


Q. If a, b, c and d are four positive real numbers such that abcd = 1, what is the minimum value of (1 + a)(1 + b)(1 + c)(1 + d)?


  1. 4

  2. 1

  3. 16

  4. 18


Ans . C


Q. For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence. If the difference in squares of 7th and 6th terms of this sequence is 517, what is the 10th term of this sequence?


  1. 147

  2. 76

  3. 123

  4. Cannot be determined


Ans . C


Q. Let x and y be two positive numbers such that x + y = 1. Then the minimum value of (x + 1/x)^2 + (y + 1/y)^2 is


  1. 12

  2. 20

  3. 12.5

  4. 13.3


Ans . C


Q. Let b be a positive integer and a = b2 – b. If b ≥ 4, then a2 – 2a is divisible by


  1. 15

  2. 20

  3. 24

  4. all


Ans . D


Q. Ujakar and Keshab attempted to solve a quadratic equation. Ujakar made a mistake in writing down the constant term. He ended up with the roots (4, 3). Keshab made a mistake in writing down the coefficient of x. He got the roots as (3, 2). What will be the exact roots of the original quadratic equation?


  1. (6, 1)

  2. (–3, –4)

  3. (4, 3)

  4. (–4, –3)


Ans . A


Q. Let n be the number of different five-digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?


  1. 144

  2. 168

  3. 192

  4. None of these


Ans . C


Quiz

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