¶ Numbers
Number theory is concerned with the properties of numbers in general, and in particular integers. As this is a huge issue we decided to divide it into smaller topics.
Video on “Numbers”:
Types of Numbers
Real Number
A real number is a value that represents any quantity along a number line. Because they lie on a number line, their size can be compared. You can say one is greater or less than another, and do arithmetic with them. In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. It can be extended infinitely in any direction and is usually represented horizontally. The numbers on the number line increase as one moves from left to right and decrease on moving from right to left.
Real numbers are divided into two categories:
● Rational numbers
● Irrational numbers
Rational Number
A rational number is a number that can be in the form p/q where p and q are integers and q is not equal to zero (Just remember! “q” cannot be equal to zero). For example -
Irrational Number
The opposite of rational numbers is irrational numbers. In simple terms, irrational numbers are real numbers that can’t be written as a simple fraction like p/q.
Take π.
π is a real number. But it’s also an irrational number because you can’t write π as a simple fraction:
π = 3.1415926535897932384626433832795 (and counting)
There’s no way to write π as a simple fraction, so it’s irrational.
The same goes for √2.
The √2 equals 1.4142135623730950... (etc).
You can’t make √2 into a simple fraction, so it’s an irrational number.
Integers
Integers are defined as: all negative natural numbers {...,−4,−3,−2,−1}, zero {0}, and positive natural numbers {1,2,3,4,...}. Note that integers do not include decimals or fractions, just whole numbers.
Video on "Properties of Integers":
Introduction to Fractions
A fraction is expressed as one number over another number. The number on top is called the numerator and the number at the bottom is called the denominator. Fractions are of three types:
● Proper fractions:
A fraction where the numerator (the top number) is less than the denominator (the bottom number). Example: 1/4 (one quarter) and 5/6 (five sixths) are proper fractions. The absolute values of proper fractions are greater than zero and smaller than 1. So, if x is 5/6 of y, then y>x.
● Improper fraction:
A fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). So it is usually "top-heavy". Example: 5/3 (five thirds) and 9/8 (nine eighths) are improper fractions. The absolute values of improper fractions are more than 1. So, if x is 9/8 of y, then x>y.
● Mixed fraction:
A whole number and a fraction are combined into one "mixed" fraction. Example: 1 1⁄2 (one and a half) is a mixed fraction. Here 1 is an integer and 1⁄2 is a fraction. 1+1⁄2 = 3/2. So mixed fraction is a different way of representing improper fractions. Like improper fractions, the absolute value of a mixed fraction is always more than 1.
Note: We get the integer part of the improper fraction by dividing the numerator by the denominator and discarding the indiscrete part. For example, in the case of 15/4, we know that 4 goes into 15 three times with 3 as leftover or reminder. So, 15/4 = 3 3⁄4. This converts the improper fraction into a mixed fraction.
Properties of Number Line
A number line is a picture of a graduated straight line that serves as an abstraction for real numbers, denoted by R. Every point of a number line is assumed to correspond to a real number and every real number to a point.
To draw a number line we draw a line with several vertical dashes in it and ordered numbers below the line, both positive and negative. The number corresponding to the point on the number line is called the coordinate of the number line.
Properties of numbers between zero and one
● 0.1, 0.5, 0.66 etc are numbers between 0 and 1
● Any number between zero and one, when raised to the power of an integer greater than one, becomes smaller than the original number i.e if 0<x<1, x2 < x. For example: (0.1)2 = 0.01 and 0.01 < 0.1.
● Any number between zero and one, when represented as a fraction, is always a proper fraction.
Example:
1. If x is greater than zero, but less than 1, which of the following is the largest?
A. x4 B. x3 C. x D. 1/x4 E. 1/x3
Answer: (D)
Explanation:
Any number between zero and one, when raised to the power of an integer greater than one, becomes smaller than the original number. So, x4 is the smallest among x4, x3, x2, and x. So the reciprocal of x4 will be the greatest.
Properties of numbers between -1 and 0:
● -0.3, -0.2, -0.88 etc are numbers between -1 and 0.
● If the index of any negative number is an even integer, it becomes a positive number. So the number becomes greater.
● If a number between -1 and zero, is raised to a power of an odd number, the numerical value increases. For example, (-0.5)3 = (-0.125) and (-0.125) > (-0.5)
Example:
● If X is greater than -1, but less than 0, which of the following is the largest?
A. x4 B. x3 C. x D. 1/x4 E. 1/x3
Answer: (D)
Explanation:
If the index of any negative number is an even integer, it becomes a positive number. So the number becomes greater. So, x4 and x2 are positive. And x4< x2. So, the reciprocal of x4 will be the greatest.
Positive numbers, negative numbers, and zero
Positive Numbers:
Positive numbers are numbers that are greater than zero. Numbers can be positive, negative or zero. Zero is neither positive nor negative. Positive numbers are the ones you most encounter in everyday life, such as 34, 9.22, etc. When shown on a number line, they are the ones drawn on the right of zero, getting larger as you move to the right.
Zero:
Zero is considered the middle point of the number line. Measurements one way are positive, and they are negative the other way.
Zero is neither positive nor negative. It is considered a Non-negative number.
Negative Numbers:
Negative numbers are numbers that are less than zero.
When you first encounter negative numbers they can be perplexing. How can a bowl contain less than zero oranges? In fact, the counting numbers cannot be negative for this reason. It makes no sense. But when we use scalar numbers, we are measuring something like temperature or height, negative values are useful.
Even and Odd Numbers
An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder. An even number is an integer of the form n=2k, where k is an integer. An odd number is an integer that is not evenly divisible by 2. An odd number is an integer of the form n=2k+1, where k is an integer.
Addition / Subtraction: even +/- even = even; even +/- odd = odd; odd +/- odd = even.
Multiplication: even * even = even; even * odd = even; odd * odd = odd.
Exponents: even ^ even=even; even ^ odd =even; odd ^ even =odd; odd ^ odd =odd.
Division: even/even= even or odd; even/odd= even; odd/odd= odd
Practice:
1. If n is an odd number, then which of the following best describes the number represented by n2+ 2n + 1?
A. It can be odd or even.
B. It must be odd.
C. It must be divisible by four.
D. It must be divisible by six.
E. Cannot be determined
2. If P is an even number, and Q and R are both odd, which of the following must be true?
A. PQ is an odd number.
B. Q-R is an even number.
C. PQ - PR is an odd number
D. Q+ R cannot equal P
E. None of these
Place Value And Face Value
The difference between place value and face value is that the place value deals with the position of the digit, and the face value represents the actual value of a digit.
The Face value of 6 in 80,156 = 6.
The Place value of 6 in 80,156 = 6×1 = 6.
The Face value of 5 in 80,156 = 5.
The Place value of 5 in 80,156 = 5×10 = 50.
The Face value of 1 in 80,156 = 1.
The Place value of 1 in 80,156 = 1× 100 = 100.
The Face value of 0 in 80,156 = 0.
The Place value of 0 in 80,156 = 0× 1,000 = 0.
Prime Numbers
A Prime number is a natural number with exactly two distinct natural number divisors: 1 and itself. Otherwise, a number is called a composite number. Therefore, 1 is not a prime, since it only has one divisor, namely 1.
>Only positive numbers can be primes.
>There are infinitely many prime numbers.
>The only even prime number is 2 since any larger even number is divisible by 2. Also, 2 is the smallest prime.
>Prime factorization: Every positive integer greater than 1 can be written as a product of one or more prime integers in a unique way. For instance, integer n with three unique prime factors a, b, and c can be expressed as n=ap∗bq∗cr, where p, q, and r are powers of a, b, and c, respectively and are ≥1.Example: 4200=23∗3∗52∗7.
>To determine if a number is a prime, follow these steps:
1. Determine the rough approximate square root of that number.
2. Divide the number by all the primes less than the approximate square root.
3. If the number is not divisible by any of the primes, it is a prime.
>Easy way to remember the number of primes from 1 to 100 is 4, 4, 2, 2, 3, 2, 2, 3, 2, 1. This signifies the numbers of primes in each tenth i.e. there are 4 prime numbers in the first 10 numbers (1-10) and 4 prime numbers from (11-20), 2 prime numbers from (21-30), and so on.
Practice:
1. How many prime numbers are there between 65 and 100?
A. 6 B. 7 C. 8 D. 9 E. None of these
2. Which of the following numbers is a prime number?
A. 147 B. 149 C. 153 D. 155 E. 161
Video on “Prime Numbers”:
Divisibility
• A number is divisible by 2 when it is even or ends in 0, 2, 4, 6, or 8.
• A number is divisible by 3 if the sum of its digits is a multiple of 3.
• A number is divisible by 4 when its last two digits are zeros or they are a multiple of 4.
• A number is divisible by 5 when it ends in 0 or 5.
• A number is divisible by 6 when it is divisible by 2 and by 3 simultaneously.
• A number is divisible by 7 when separating the first digit on the right, multiplying by 2, subtracting this product from what is left, and so on, gives zero or a multiple of 7.
• A number is divisible by 8 when its last three digits are zeros or they are a multiple of 8.
• A number is divisible by 9 when the sum of its digits is a multiple of 9.
• A number is divisible by 10 when it ends in 0.
• A number is divisible by 11 when the difference between the sum of the absolute values of odd numbers place and the sum of the absolute values of torque figures place from right to left, is zero or a multiple of 11.
• A number is divisible by 25 when its last two digits are zeros or they are a multiple of 25.
• A number is divisible by 125 when its last three digits are zeros or are a multiple of 125.
Practice:
1. (xn– an) is completely divisible by (x - a), when:
A. n is any natural number
B. n is an even natural number
C. n is an odd natural number
D. n is prime
E. n is a negative integer
2. The largest natural number which exactly divides the product of any four consecutive natural numbers is:
A. 6 B. 12 C. 24 D. 120 E. Cannot be determined
3. Which of the following numbers is divisible by 11?
A. 30,217
B. 44,221
C. 67,523
D. 70,711
E. None of these
4. Which of the following numbers is divisible by 36?
A. 35,925
B. 72,340
C. 74,098
D. 152,640
E. 192,042
Video on “Divisibility”:
Factors
A divisor of an integer n, also called a factor of n, is an integer that evenly divides n without leaving a remainder.
>Every integer is a divisor of 0, except, by convention, 0 itself.
To factor a number means to break it up into numbers that can be multiplied together to get the original number. Example: 6 = 3 x 2 so, factors of 6 are 3 and 2 9 = 3 x 3 so, factors of 9 are 3 and 3.
Sometimes, numbers can be factored into different combinations.
A number can have many factors.
For example: What are the factors of 12?
● 3 × 4 = 12, so 3 and 4 are factors of 12
● 2 × 6 = 12, so 2 and 6 are also factors of 12
● and 1 × 12 = 12, so 1 and 12 are factors of 12 as well
So 1, 2, 3, 4, 6 and 12 are all factors of 12 And -1, -2, -3, -4, -6 and -12 also, because multiplying negatives makes a positive.
Example:
1. 240 students in a group are to be seated in rows so that there are an equal number of students in each row. Each of the following could be the number of rows EXCEPT.
A. 4 B. 20 C. 30 D. 40 E. 90
Answer: Option (E)
Explanation:
240/90 is not an integer. Because 90 is not a factor of 240. So, the answer is 90
2. If x and y are two distinct positive integers divisible by 2, then which of the following is necessarily divisible by 4? .
A. x + y B. x – y C. x2+ y2 D. 2x + y E. none of these
Answer: C
3. Find the value of Z if (x+1) is a factor of x3+ Zx + 3x2– 2.
A. 6 B. 5 C. 4.5 D. 0 E. none of these
Answer: D
Multiples
In math, the meaning of a multiple is the product result of one number multiplied by another number (The result of multiplying a number by an integer, not by a fraction). Here, 56 is a multiple of 7.
For example -
● 12 is a multiple of 3, because 3 × 4 = 12
● −6 is a multiple of 3, because 3 × −2 = −6
● But 7 is NOT a multiple of 3
Example:
1. 3 and 7 are factors of F. From this information, we can conclude that:
A. 8 is a factor of F
B. F is a multiple of 21
C. F=3 × 7
D. 21 is a multiple of 21
E. 3 and 7 are the only factors of F
Answer: Option (B)
Explanation:
Since 21 is divisible by both 3 and 7, F is a multiple of 21.
Highest Common Factor- HCF
The greatest common divisor (GCD), also known as the greatest common factor (GCF), or highest common factor (HCF), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. To find the HCF, you will need to do prime-factorization. Then, multiply the common factors (pick the lowest power of the common factors).
>Every common divisor of a and b is a divisor of HCF(a, b)
Lowest Common Multiple - LCM
The lowest common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then lcm(a, b) is defined to be zero.
To find the LCM, you will need to do prime-factorization. Then multiply all the factors (pick the highest power of the common factors).
>a*b=HCF(a, b)*LCM(a, b).
>H.C.F. and L.C.M. of Fractions
LCM of fractions = LCM of numerators/ HCF of denominators
HCF of fractions = HCF of numerators/ LCM of denominators
Example:
01. What is the smallest number of apples that can distributed equally among 6, 9, 15, 18 students having a surplus of 3 apples each time?
A. 422 B. 362 C. 183 D. 62 E. none of these
Answer: Option (C)
Explanation:
LCM of 6, 9, 15 & 18= 90.
Option (A) 422 divided by 90 leaves a remainder of 62. So, it can be eliminated.
Option (B) 362 divided by 90 leaves a remainder of 2. So, it can be eliminated.
Option (C) 183 divided by 90 leaves a remainder of 3. So, it can be a possible option.
Option (D) 62 divided by 90 leaves a remainder of 62. So, it can be eliminated.
Only Option (C) matches the condition. So, Option (E) can be eliminated.
So, the answer is Option (C) 183.
02. What is the smallest possible number that leaves a remainder of 1 when divided by 3, 4, 5, and 7?
A. 141 B. 106 C. 421 D. 420 E. 85
Answer: C
03. What is the least number which when divided by 3, 4 and 5 leaves remainders of 1, 2 and 3 respectively?
A. 59 B. 58 C. 116 D. 118 E. 67
Answer: B
04. If x, y, z are positive integers and 4x = 3y = 5z, then what is the smallest value of x + y + z?
A. 12 B. 47 C. 60 D. 94 E. Cannot be determined
Answer: B
05. Rafin has 18 apples, 27 mangoes and 54 bananas. He wants to divide each kind of fruit equally among the same number of children so that no fruit remains left. What is the maximum number of children who might get the fruit?
A. 3 B. 6 C. 9 D. 18 E. None of these
Answer: C
06. What is the minimum number of apples that must be added to the existing stock of 770 chocolates so that the total stock can equally be divided among 8, 12 or 16 persons?
A. 32 B. 46 C. 58 D. 80 E. none of these
Answer: B
07. If x/2, x/3 and x/13 represent integers, then x can be
A. 42 B. 56 C. 70 D. 78 E. 126
Answer: 78
Factorials
Factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For instance 5!=1∗2∗3∗4∗5.
>Note: 0!=1.
>Note: factorial of a negative number is undefined.
Fractions
When two fractions are given, there is a simple trick to help figure out which fraction is larger. Multiply the numerator of the first fraction (the top number in the fraction) by the denominator of the second fraction (the bottom number in the fraction). Again, multiply the numerator of the second fraction by the denominator of the first fraction. Then compare the two answers. Let’s compare 3/8 and 4/9.
• 3 * 9 = 27 • 4 * 8 = 32 Since, 32 > 27 We can conclude, 4/9 > 3/8.
Example:
1. Which of the fractions is the largest?
A. 14/15 B. 9/17 C. 5/6 D. 17/23 E. 19/27
Answer: A
2. A club has equal number of male and female members. On a certain day, two thirds of the
members were absent. Of the members present, one third was male. What is the ratio of male
and female who were not present on that day?
A. 1/3 B.2/3 C. 3/5 D. 7/5 E. none of these
Answer: D
3. Of the animals in Dhaka Zoo, 1/7 are Zebras, 1/4 are Giraffes, 1/3 are Tigers, and the rest is
comprised of 46 Deer. How many animals are there in the Zoo?
A. 140 B. 180 C. 168 D. 140 E. 125
Answer: C
Consecutive Integers
Consecutive integers are integers that follow one another, without skipping any integers. 7, 8, 9, and -2, -1, 0, 1, are consecutive integers.
• The product of n consecutive integers is always divisible by n!. Given n=4 consecutive integers: {3,4,5,6}. The product of 3*4*5*6 is 360, which is divisible by 4!=24.
Decimal Representation
The decimals have ten as their base. Decimals can be terminating (ending) (such as 0.78, 0.2) or repeating (recurring) decimals (such as 0.333333....).
• To convert a terminating decimal to fraction:
1. Calculate the total numbers after decimal point
2. Remove the decimal point from the number
3. Put 1 under the denominator and annex it with "0" as many as the total in step 1
4. Reduce the fraction to its lowest terms.
Example: Convert 0.56 to a fraction.
1: Total number after the decimal point is 2.
2 and 3: 56/100.
4: Reducing it to lowest terms: 56/100=14/25.
• To convert a recurring decimal to fraction:
1. Separate the recurring number from the decimal fraction
2. Annex denominator with "9" as many times as the length of the recurring number
3. Reduce the fraction to its lowest terms.
Example #1: Convert 0.393939... to a fraction.
1: The recurring number is 39.
2: 39/99, the number 39 is of length 2 so we have added two nines.
3: Reducing it to lowest terms: 39/99=13/33.
Rounding
Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep. Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.
Example:
1. The number 89.783 rounded off to the nearest tenth is equal to which of the following?
A. 90.0 B. 89.78 C. 89.7 D. 89.8 E. 89.9
Answer: D
2. Which of the following results if 65135 is rounded off to 2nd significant figure?
A. 65100 B. 65000 C. 65130
D. 66000 E. None of these
Answer: B
3. What is the product of 23 and 79 to one place of accuracy?
A. 1600 B. 1817 C. 1000
D. 1800 E. 2000
Answer: E
Video on “Rounding Off”:
Video On “Approximate problems”:
EXPONENTS
Exponents are a "shortcut" method of showing a number that was multiplied by itself several times. For instance, number a multiplied n times can be written as an, where a represents the base, the number that is multiplied by itself n times and n represents the exponent. The exponent indicates how many times to multiply the base, a, by itself.
Exponents one and zero: a0=1 Any nonzero number to the power of 0 is 1. For example 50=1and (−3)0=1.
a1=a, Any number to the power 1 is itself.
Powers of zero: If the exponent is positive, the power of zero is zero: 0n=0, where n>0.
If the exponent is negative, the power of zero (0n, where n<0) is undefined because division by zero is implied.
Powers of one: 1n=1 The integer powers of one are one.
Negative powers: a-n=1/an
Powers of minus one: If n is an even integer, then (−1)n=1. If n is an odd integer, then (−1)n=−1.
Operations involving the same exponents: Keep the exponent, multiply or divide the bases
an∗bn=(ab)n
an / bn=(a/b)n
(am)n=amn
Operations involving the same bases: Keep the base, add or subtract the exponent (add for multiplication, subtract for division)
an∗am=an+m
an/am=an-m
Exponential Equations: When solving equations with even exponents, we must consider both positive and negative possibilities for the solutions.
For instance a2=25, the two possible solutions are 5 and −5. When solving equations with odd exponents, we'll have only one solution. For instance for a3=8, the solution is a=2 and for a3=−8, the solution is a=−2.
Example:
1. What is the greatest positive integer n such that 3n is a factor of 1814 ?
A. 10 B. 12 C. 16 D. 28 E. 60
Answer: D
2. If both 72 and 32 are factors of x where x = n × 25× 62× 113, what is the smallest possible
positive value of n?
A. 25 B. 16 C. 49 D. 75 E. none of these
Answer: C
3. What is the last digit of 6218?
A. 2 B. 4 C. 6 D. 8 E. none of these
Answer: C
4. What is the last digit of 4218?
A. 2 B. 4 C. 6 D. 8 E. None of these
Answer: C
5. If x ≠ 0, then (x3)2*x2=
A. 1 B. x2 C. x8 D. x11 E. x12
Answer: C
6.(√3 + √2)/(√3 - √2)=
A. 5 + 2√6 B. (5 + 2√6)/5
C. 5 - 2√6 D. (5 - 2√6)/5
E. None of these
Answer: A
7. 210 + 210 + 210 + 210 =?
A. 20000 B. 240 C. 211 D. 212 E.210000
Answer: D
8. If (0.063*3x+y)/(0.007 x 9y)= 1, then x-y=?
A. 0 B. 1 C. 2 D. -1 E. -2
Answer: E
Video on “Exponents”:
ORDER OF OPERATIONS - PEMDAS
Perform the operations inside a Parenthesis first (absolute value signs also fall into this category), then Exponents, then Multiplication and Division, from left to right, then Addition and Subtraction, from left to right - PEMDAS.
Video on the basics of “Number”:
Video on "Prime Numbers":
Video on "Remainders":