Number SystemThe numbers can be defined in a lot of different ways like positive, negative, even, odd, natural, whole, integers, fractions, etc.

This chapter deals with all these i.e different kinds of numbers.

Positive/Negative

Numbers can either be positive or negative or even none of the these!!

| | | | | | | | | | |

-5 -4 -3 -2 -1 0 1 2 3 4 5

A number line illustrates this.

A negative number is defined as any number left to the zero or a number less than zero. The symbol used to denote a negative number is −.

A positive number is defined as any number right to the zero or a number more than zero. The symbol used to denote a positive number is +.

As we can see that the number zero is just used to make a distinction between positive and negative number, so it is considered to be neither positive or negative i.e zero (0) is a neutral number.

Things to keep in mind:-

1. Positive × positive = positive2. positive × negative = negative Multiplication3. negative × negative = positive

4. positive/positive = positive 5. positive/negative = negative Division6. negative/negative = positive7. A double negative means positive. For example:- 4 – (-2) = 4 + 2= 6

A few more definitions:-

1. Natural numbers:- 1, 2, 3, 4, 5, ………..(only positive)2. Whole numbers:- 0, 1, 2, 3, ………………(non – negative)3. Integers:-………………. -5, -4, -3, -2, -1, 0, 1, 2, 3……………(negative/positive)4. Fractions:- numbers of the form p/q where q ≠ 0

Even/Odd

Even:- An even number is an integer that is divisible by 2. For example:- -24, -36, -20, 0, 20, 42, 38 etc. An even number can be written in the form n = 2k, where k is an integer.

Odd:- An odd number is an integer that is not divisible by 2. For example:- -23, -37, -19, 1, 3, 17, etc. An odd number can be written in the form n = 2k+1, where k is an integer.

Things to remember:-

1. even + even = even2. even + odd = odd Addition3. odd + odd = even4. even – even = even5. even – odd = odd Subtraction6. odd – odd = even7. even × even = even8. even × odd = even Multiplication

9. odd × odd = odd10. Division of even or odd numbers does not follow any specific rules. It may result in an even or odd integer or a fraction. For

example: - 6/2 = 3, 6/3 = 2, 35/5 = 7, 6/4 ≠ integer11. The only specific rule for division is (Odd/Even) ≠ integer i.e an odd integer when divided by an even integer would never

result in an integer.

Consecutive Integers:-

The word consecutive means one after the other. Similarly, consecutive numbers are the numbers that follow one another from a given value.

For Example:- 1, 2, 3, 4 are consecutive integers and

-12, -13, -14, -15, 16 are also consecutive integers.

Consecutive integers can also make some specific patterns like:-

Consecutive even integers:- 2, 4, 6, 8, 10……… Consecutive odd integers:- 1, 3, 5, 7, 9…… Consecutive multiples of 5:- 5, 10, 15, 20, 25………..

Things to remember:-

1. The arithmetic mean (average) is equal to the median in a set of consecutive numbers.2. The average and the median are both equal to the average of the 1st and last numbers of the set.

For Example:- In the set 2, 4, 6, 8……………..200, the average and the median are both equal to the average of the 1st and the last numbers i.e Average = median = (2+200)/2= 101

3. Two consecutive integers are never divisible by the same prime number and therefore by the same number.

4. To count the number of integers from a to b, subtract a from b and add 1 to the result.For example:- The number of integers from 2 to 7 is not 5 but 6 (2, 3, ,4, 5, 6 and 7)But the number of integers between 2 and 7 is 4.

5. The product of n consecutive numbers is always divisible by n.6. The sum of n consecutive integers is always divisible by n if n is odd and never divisible by n if n is even.

For example:- The sum of 1, 2 and 3 i.e 6 is divisible by 3(number of integers is odd) but he sum of 1, 2, 3 and 4 i.e 10 is not divisible by 4(number of integers is even)

Divisibility Rules:-

A few important divisibility rules:-

2 - A number is divisible by 2 if the last digit of the number is even

3- A number is divisible by 3 if the sum of the digits of the number is divisible by 3

4- A number is divisible by 4 if the last two digits form a number that is divisible by 4

5- A number is divisible by 5 if the last digit is 0 or 5

6- A number is divisible by 6 if the number is divisible by 2 and 3

7- Take the last digit, double it and subtract from the rest of the number. If the answer is divisible by 7, then the number is divisible by 7.

For example:- To check whether 343 is divisible by 7. We double the last digit i.e 3 × 2 = 6 and subtract it from the rest of the number i.e 34. We get 34 – 6 = 28. The result 28 is divisible by 7 so is the original number i.e 343 is divisible by 7

8- A number is divisible by 8 if the last three digits form a number that is divisible by 8

9- A number is divisible by 9 if the sum of the digits of the number is divisible by 9

10 - A number is divisible by 10 if the number ends in a 0

12- A number is divisible by if the number is divisible by both 3 and 4

25 - A number is divisible by 25 if the number ends in 00, 25, 50 or 75

Prime numbers:-

A prime number is a positive integer which has exactly two factors i.e 1 and the number itself.

Things to remember:-

1. 1 is not prime.

2. The smallest prime number is 2

3. The only even prime number is 2

4. All prime numbers except 2 and 5 end in 1, 3, 7 or 9

Factors/Multiples:-

A factor is a positive integer that divides evenly into an integer. In general, it is said ‘m’ is a factor of ‘n’, for non-zero integers m and n, if there exists a relation such that n/m = k, where k is an integer.

A multiple is an integer that can be evenly divided into an integer. In general, it is said that ‘m’ is a multiple of ‘n’, for non-zero integers m and n, if there exists a relation such that m = nk, where k is an integer.

Things to remember:-

1. 1 is a factor of every integer

2. Every integer is a factor and a multiple of itself. It is the smallest positive multiple of itself and the largest positive factor of itself.

3. If ‘x’ is a factor of ‘y’ and ‘y’ is a factor of ‘z’, then ‘x’ is a factor of ‘z’

4. If ‘x’ is a factor of ‘y’ and ‘x’ is a factor of ‘z’, then ‘x’ would be a factor of (y+z)

5. If ‘x’ is a factor of ‘y’ and ‘y’ is a factor of x, then x = y or x = -y

6. All numbers have a limited number of factors and an unlimited number of multiples

Number of factors:-

As we discussed, all the numbers have a limited number of factors so we can be asked to find the numbers of factors of a number.

If you are asked to find the number of factors of an integer, follow the below mentioned steps:-

1) Make the prime factorization of the integer i.e write the integer in the form n = ap × bq × cr…….. , where a, b, c are the prime factors of n and p, q, r are their powers.

2) The number of factors of n will be given by (p+1)(q+1)(r+1)……….

Example:- What is the number of factors of 441?

Sol:- 15435 = 32 × 51 × 73

So, the number of factors will be (2+1)(1+1)(3+1) = 3 × 2 × 4 = 24

Remainder:

Remainder is the left over number when we divide two numbers which do not divide evenly into each other.

When we divide 23 by 5, it does not divide evenly and 3 is left over. This left over is known as the remainder. So, when 23 is divided by 5, 3 is the remainder.

A remainder is always less than the divisor.

4 Quotient

Divisor 5 23 Dividend

20

3 Remainder

The relation between a divisor, dividend and remainder is given by

Dividend = (Divisor) (Quotient) + Remainder

23 = (5) (4) + 3

If y when divided by x leaves a remainder of r1

and z when divided by x leaves a remainder of r2

if (y + z) is divided by x, it would leave a remainder of r1 + r2

(|y – z|) when divided by x will leave a remainder of |r1 – r2|

(y) (z) when divided by x will leave a remainder r1r2 .

HCF or GCF,LCM

HCF or GCM or GCD Greatest Common Factor (Divisor)

The GCF of two or more non-zero integers, is the largest positive integer that divides the numbers evenly or without remainder.

Example Find the GCF of 42 and 54.

Solution 42 = 2 X 3 X 7

54 = 2 X 3 X 3 X 3

To find the LCM, pick all the primes (with maximum power available).

In this case, 2 X 7 X 33 = 378 is the LCM.

Note :- If two numbers have NO primes in common, than their GCF is 1 and the LCM is simply their product.

For example – The GCF of 14 (7 X 2) and 55 (5 X 11) is 1

and the LCM is 14 X 55 = 770.

The product of LCM and GCF of two integers = the product of the numbers.

Perfect Square :-

A perfect square is a number obtained by multiplying a number by itself. For example, 4 = 2 X 2, 9 = 3 X 3, 16 = 4 X 4.

Facts about perfect squares :-

1. The number of distinct factors of a perfect square is always odd.

2. A perfect square can never have its units digits as 2, 3, 7 or 8.

3. A perfect square always has an odd number of odd factors and even number of even factors.

4. The sum of distinct factors of a perfect square is always odd.

Factorial :-

Factorial of a positive integer, denote by n! , is the product of all positive integers less than or equal to n. for example :-

6! = 6 X 5 X 4 X 3 X 2 X 1

Things to remember :-

1. 0! = 1

2. n! is divisible by all the integers from 1 to n.

3. factorials cannot be factorized i.e.,

10! ≠ 5! X 2!

Practice Questions:-

Q1. If a, b, c and d are prime numbers such that 1 < a < b < c < d and abcd = 1430, then what is the value of d?

A) 3B) 9C) 11D) 13E) 22

Q2. What is the least positive integer that is divisible by each of the integers 1 through 7, inclusive?

A) 420B) 840C) 1260D) 2520E) 5040

Q3. If x is equal to the sum of even integers from 40 to 60 inclusive and y is the number of even integers from 40 to 60 inclusive. What is the value of x + y?

A) 550B) 551C) 560D) 561E) 572

Q4. If S is a set of four numbers w, x, y and z. Is the range of the numbers in S greater than 2?

1) w – z > 22) z is the least number in S

Q5. If a, b and c are integers, is the number 3(a + b) + c divisible by 3?

1) a + b is divisible by 32) c is divisible by 3

Q6. Is x an even integer?

1) x is the square of an integer2) x is the cube of an integer

Q7. If x and y are the integers between 10 and 99, inclusive. Is an integer?

1) x and y have the same two digits, but in reverse order.2) The tens digit of x is 2 more than the units digit and the tens digit of y is 2 less than the units digit.

Q8. How many prime factors does 540 + 537 have?

A) 1B) 2C) 3D) 4E) 5

Q9. If r, s and t are consecutive integers, what is the greatest prime factor of 3r + 3s + 3t ?

A) 3B) 5C) 7D) 11E) 13

Q10. What is the sum of digits of the number 1050 – 74?

A) 9B) 180C) 275D) 360E) 440

Q11. If x, y and z are positive integers such that ‘x’ is a factor of’ y’ and x is a multiple of ‘z’. Which of the following is NOT necessarily an integer?

A)B)C)D)

E)Q12. Is the integer z divisible by 6?

1) The greatest common factor of z and 12 is 3.2) The greatest common factor of z and 15 is 15.

Q13. If x and y are perfect squares, then which of the following is not necessarily a perfect square?

A) x2

B) xyC) 4xD) x +yE) x5

Q14. If p and q are positive integers, how many integers are larger than pq and smaller than p(q + 2)?

A) 3B) P + 2C) p – 2D) 2p – 1E) 2p + 1

Q15. If x > y > 0, which of the following must be true?

I.II.III.A) I onlyB) II onlyC) III onlyD) I and II onlyE) II and III only

Q16. If 2 is the greatest number that will divide evenly into both x and y, what is the greatest number that will divide evenly into both 5x and 5y?

A) 2B) 4C) 6D) 8E) 10

Q17. If p divided by 9 leaves a remainder of 1, which of the following must be true? I. p is evenII. p is oddIII. p = 3z + 1, for some integer z

A) I onlyB) II onlyC) III onlyD) I and II onlyE) None of the above

Q18. If the sum of two prime numbers x and y is odd, then the product of x and y must be divisible by

A) 2B) 3C) 4D) 5E) 8

Q19. If and x and y are integers, then which one of the following must be true?

A) x is divisible by 4B) y is oddC) y is evenD) x is evenE) x is irreducible fraction

Q20. If y is an odd integer and the product of x and y equals 222, what is the value of x?

1) x is a prime number2) y is a three digit number

Q21. If x and y are prime numbers such that x > y > 2, then x2 – y2 must be divisible by which one of the following numbers?

A) 3B) 4C) 5D) 9E) 12

Q22. If the positive integer N is a perfect square, which of the following must be true?I. Number of distinct factors of N is oddII. Sum of distinct factors of N is oddIII. The number of distinct prime factors of N is even

A) I onlyB) II onlyC) I and II onlyD) I and III onlyE) I, II and III

Q23. In a certain game, a large bag is filled with blue, green, purple and red chips worth 1, 5, x and 11 points each, respectively. The purple chips are worth more than the green chips but less than the red chips. A certain number of chips are then selected from the bag. If the product of the point values of selected chips is 88,000, how many purple chips were selected?

A) 1B) 2C) 3D) 4E) 5

Q24. Is the positive integer x a perfect square?

1) The number of distinct factors of x is even2) The sum of all distinct factors of x is even

Q25. Let a = sum of integers from 1 to 20 and b = sum of integers from 21 to 40. What is the value of b – a ?

A) 21B) 39C) 200D) 320E) 400

Q26. Does x – y = 0?

1)2) x2 = y2

Q27. If p and q are integers, is pq + 1 even?

1) If p is divided by 2, the remainder is 12) If q is divided by 6, the remainder is 1

Q28. S is a set of integers such thatI. If x is in S, then –x is in SII. If both x and y are in S, then so is x + y.

Is -2 in S?

1) 1 is in S2) 0 is in S

Q29. A botanist select n2 trees on an island and studies (2n + 1) trees everyday where n is an even integer. He does not study the same tree twice. Which of the following cannot be the number of trees that he studies on the last day?

A) 13B) 17C) 28D) 31E) 79

Q30. If x = 2891 × 2892 × 2893 ×……………..× 2898 × 2899 × 2900, the what is the remainder when x is divided by 17?

A) 0B) 4C) 7D) 10E) 14

Q31. For every positive integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest factor of h(100) + 1, then p is

A) Between 2 and 10B) Between 11 and 20C) Between 21 and 30D) Between 31 and 40E) Greater than 40

Q32. If the sum of five consecutive positive integers is A, then the sum of the next five consecutive integers in terms of A is

A) A + 1B) A + 5C) A + 25D) 2AE) 5A

Q33. A, B, C and D are all different digits between 0 and 9. If AB + DC = 7B (AB, DC and 7B are two digit numbers), what is the value of C?

A) 0B) 1C) 2D) 3E) 5

Q34. If 2x < y and 2x2 > xy, which of the following must be true?

A) x > 1

B) x < 0C) x > 0D) y < 0E) y > 1

Q35. If x and y are different positive integers and 3x + y = 14, what is the product of all the possible values of x?

A) 6B) 8C) 14D) 20E) 24

Q36. If x is an integer and , which of the following must be true?

I. a is evenII. a is positiveIII. a is an integer

A) I onlyB) II onlyC) III onlyD) I and II onlyE) None of the above

Q37. Which of the following expressions has the greatest value? (numbers)(650-750)

A)B)

C)D)E)

Q38. If ‘m’ and ‘n’ are integers then what is the smallest possible value of integer m such that 0.3636363636……..?

A) 3B) 4C) 7D) 13E) 22

Q39. If N = 1234@ and @ represents the units digit. Is N a multiple of 5? (numbers)(600-700)

1) @! is not divisible by 52) @ is divisible by 9

Q40. If (|p|!)p = |p|!, which of the following could be the value(s) of p?

A) -1B) 0C) 1D) -1 and 1E) -1, 0 and 1

Q41. If 5x = y +7, is (x – y) > 0?

1) xy = 62) x and y are consecutive integers with the same sign

Q42. If ‘s’ and ‘t’ are positive integer such that , which of the following could be the remainder when ‘s’ is divided by ‘t’?

A) 2B) 4C) 8D) 20E) 45

Q43. If ‘x’ is positive, which of the following could be the correct ordering of , 2x and x2?

I. x2 < 2x <

II. x2 < < 2x

III. 2x < x2 <

A) None of the aboveB) I onlyC) III onlyD) ;I and II onlyE) II and III only

Q44. If p is a prime number greater than 2, what is the value of p?

1) There are total of 100 prime numbers between 1 and p + 1.2) There are a total of ‘p’ prime numbers between 1 and 3912

Q45. If a, b and c are positive integers and are assembled into the six digit number abcabc, which of the following must be a factor of abcabc?

A) 16B) 13C) 5D) 3E) None of the above

Q46. What is the remainder when 11 + 22 + 33 +………………..+ 1010 is divided by 5?

A) 0B) 1C) 2D) 3E) 4

Q47. If Q and T are integers, what is the value of Q?

1)2)

Q48. If ‘m’ and ‘n’ are integers and , which of the following cannot be a value of m + n?

A) 25B) 29C) 50D) 52

E) 101

Q49. The three digits of a number add upto 11. The number is divisible by 5. The leftmost digit is double the middle digit. What is the product of the three digits?

A) 40B) 72C) 78D) 88E) 125

Q50. If x = y + y2 and y is a negative integer, when y decreases in value, then x

A) Increases in valueB) FluctuatesC) Decreases in valueD) Remains the sameE) Decreases in constant increments

Q51. If w, x, y and z are the digits of the four digit number N, a positive integer, what is the remainder when N is divided by 9?

1) w + x + y + z = 132) N + 5 is divisible by 9

Q52. If both ‘x’ and ‘y’ are positive integers less than 100 and greater than 10, is the sum ‘x + y’ a multiple of 11?

1) (x – y) is a multiple of 222) The tens digit and the units digit of ‘x’ are the same; the tens digit and units digit of ‘y’ are the same

Q53. Susie can buy apples from two stores: a supermarket that sells apples only in bundles of 4 and a convience store that sells single, unbundled apples. If Susie wants to ensure that the total number of apples she buys is a multiple of 5, then what is the minimum number of apples she must buy from the convience store?

A) 0B) 1C) 2D) 3E) 4

Q54. ‘x’ is the sum of ‘y’ consecutive integers, ‘w’ is the sum of ‘z’ consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A) x = wB) x > wC) (x/y) is an integerD) (w/z) is an integerE) (x/z) is an integer

Q55. If ‘x’ is a positive integer, is ‘x’ prime? (prime numbers) (700-800)

1) ‘x’ has the same number of factors as y2, where y is a positive integer greater than 22) ‘x’ has the same number of factors as ‘z’, where ‘z’ is a positive integer greater than 2

Q56. The greatest common factor of 16 and the positive integer ‘n’ is 4 and the greatest common factor of ‘n’ and 45 is 3. Which of the following could be the greatest common factor of ‘n’ and 10?

A) 3B) 14C) 30D) 42E) 70

Q57. If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest prime factor of k?

A) 5B) 7

C) 11D) 13E) 17

Q58. If ‘a’ and ‘b’ are positive integers such that , which of the following must be a divisor of ‘a’?

A) 10B) 13C) 18D) 26E) 50

Q59. If ‘a’ and ‘b’ are positive integers divisible by 6, is 6 the greatest common divisor of ‘a’ and ‘b’?

1) a = 2b + 62) a = 3b

Q60. Is x < z?

1) x = y

2) y = 0.5 z

Q61. If x and y are integers such that x < y < o, what is the value of x –y?

1) (x + y) (x – y) = 52) xy = 6