Type Of Numbers:-
The number theory or number systems is the back bone for CAT preparation. Number system not only form the basis of most calculations and other systems in mathematics, but also it forms a major percentage of the CAT quantitative section. The reason for that is the ability of examiner to formulate tough conceptual questions and puzzles from this section. In number systems there are hundreds of concepts and variations, along with various logic attached to them, which makes this topic more complex in preparation for the CAT examination. The students while going through these topics should be careful in capturing the concept correctly, as it’s not the speed but the concept that will solve the question here. The correct understanding of concept is the only way to solve complex questions based on this section.
Real numbers: These numbers that can represent the physical quantities in a complete manner. All real numbers can be measured and can be represented on a number line. These are of two types:
Rational numbers: A number that can be represented in the form p/q where p and q are integers and q is not zero. Example: 2/3, 1/10, 8/3 etc. They can be finite decimal numbers, whole numbers, integers, fractions.
Irrational numbers: A number that cannot be represented in the form p/q where p and q are integers and q is not zero. An infinite non recurring decimal is an irrational number. Example: √2, √5 , √7 and Π(pie)=3.1416.
The rational numbers are classified into Integers and fractions.
Integers: The set of numbers on the number line, with the natural numbers, zero and the negative numbers are called integers, I = {…..-3, -2, -1, 0, 1, 2, 3…….}
Fractions:
A fraction denotes part or parts of an integer. For example 1/5, which can represent 1/5th part of the whole. The type of fractions are:
1. Common fractions: The fractions where the denominator is not 10 or a multiple of it. Example: 3/7, 6/8 etc.
2. Decimal fractions: The fractions where the denominator is 10 or a multiple of 10. Example 9/10, 3/1000 etc.
3. Proper fractions: The fractions where the numerator is less than the denominator. Example 1/7, 4/6, 7/9 etc. its value is always less than 1.
4. Improper fractions: The fractions where the numerator is greater than or equal to the denominator. Example 5/4, 6/5, 9/2 etc. Its value is always greater than or equal to 1.
5. Compound fraction: A fraction of a fraction is called a compound fraction
Example 2/3 of 4/6 = 2/3 x 4/6 = 8/18
6. Complex fractions: The combination of fractions is called a complex fraction.
Example (4/6)/ (7/8)
7. Mixed fractions: A fraction which consists of two parts, an integer and a fraction. Example 5 ½, 7 ¾
Example: Express 27/8 as a mixed fraction
Ans. Divide the numerator by denominator; note the multiplier, whatever remainder is left divide it with the original denominator. For 27/8, 24/8 = 3, and remainder left is 3, therefore 3 3/8 is the mixed fraction
Example: Express 35x7/17as an improper fraction.
Ans. Here we need to multiply the denominator with the non-fraction part and add it to numerator and using same denominator.
For 35x7/17= = 602/17
The integers are classified into negative numbers and whole numbers
Negative numbers: All the negative numbers on the number line, {…..-3, -2, -1}
Whole numbers: The set of all positive numbers and 0 are called whole numbers, W = {0, 1, 2, 3, 4…….}.
Natural numbers: The counting numbers 1, 2, 3, 4, 5……. are known as natural numbers, N = {1, 2, 3, 4, 5…..}. The natural numbers along with zero make the set of the whole numbers.
Even numbers: The numbers divisible by 2 are even numbers. e.g., 2, 4, 6,8,10 etc. Even numbers can be expressed in the form 2n where n is an integer other than 0.
Odd numbers: The numbers which are not divisible by 2 are odd numbers. e.g. 1, 3, 5, 7, 9 etc. Odd numbers are expressible in the form (2n + 1) where n is an integer other than 0.
Composite numbers: A composite number has other factors besides itself and unity .e.g. 8, 72, 39 etc. A real natural number that is not a prime number is a composite number.
Prime numbers: The numbers that has no other factors besides itself and 1 is a prime number. Example: 2, 23,5,7,11,13 etc. Here are some properties of prime numbers:
• The only even prime number is 2
• 1 is neither a prime nor a composite number
• If p is a prime number then for any whole number a, ap – a is divisible by p.
• 2,3,5,7,11,13,17,19,23,29 are first ten prime numbers (should be remembered)
• Two numbers are supposed to be co-prime of their HCF is 1, e.g. 3 & 5, 14 & 29 etc.
• A number is divisible by (ab) only when that number is divisible by each one of (a) and (b), where (a) and (b) are co prime.
• To find a prime number, check the rough square root of the given number and divide the number by all the prime number lower than the estimated square root
• All prime numbers can be expressed in the form 6n-1 or 6n+1, but all numbers that can be expressed in this form are not prime
Example: If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is: (CAT 2003)
(a) 1 (b) 2
(c) 3 (d) more than 3
Ans. (a) a, a + 2, a + 4 are prime numbers. The number fits is 1, 3, 5 and 3, 5, 7 but post this nothing will fit. Now 1, 3, 5 are not prime numbers as 1 is not prime number.
So, only one possibility is there 3, 5, 7 for a = 3.
Prime Factors: The composite numbers express in factors, wherein all the factors are prime. To get prime factors we divide number by prime numbers till the remainder is a prime number. All composite numbers can be expressed as prime factors, for example prime factors of 150 are 2,3,5,5.
A composite number can be uniquely expressed as a product of prime factors.
e.g. 12 = 2 x 6 = 2 x 2 x 3 = 22 x 31
20 = 4 x 5 = 2 x 2 x 5 = 22 x 51 etc
Note : The number of divisors of a given number N ( including one and the number itself ) where N = am x bn x cp ……. Where a, b, c are prime numbers
are = ( 1 + m ) ( 1 + n ) ( 1 + p ) …………..
e.g. 90 = 2 x 3 x 3 x 3 x 5 = 21 x 32 x 51
Hence here a = 2 b = 3 c = 5
m = 1 n = 2 p = 1
then the number of divisors are = ( 1 + m ) ( 1 + n ) ( 1 + p ) = 2 x 3 x 2 = 12
the factors of 90 = 1 , 2 , 3 , 5 , 6 , 9 , 10 , 15 , 18 , 30 , 45 , 90 = 12
the sum of divisors of given number N is
( am+1 – 1 ) ( bn+1 - 1 ) ( cp+1 – 1 ) ……..
___________________________________
( a – 1 ) ( b – 1 ) ( c – 1 ) ……….
Perfect number: If the sum of the divisor of N excluding N itself is equal to N , then N is called a perfect number. e.g. 6, 28, 496
Finding a perfect number through Euclid’s method
Euclid's method makes use of the powers of 2, which are numbers obtained by multiplying by 2 by itself over and over again, which are 1, 2, 4, 8, 16, 32, 64, 128….
Note that the sum of the two numbers in this series (in ascending order) is equal to the third number minus 1:
1+2 = 3 = 4 - 1,
1+2+4 = 7 = 8 - 1,
STARTING FROM THE NUMBER 1, IF YOU ADD THE POWERS OF 2 AND IF THE SUM IS A PRIME NUMBER, THEN YOU GET A PERFECT NUMBER BY MULTIPLYING THIS SUM TO THE LAST POWER OF 2.
If you add 1+2, the sum is 3, which is a prime number. Therefore 3 x 2 = 6 is a perfect number.
If you add 1+2+4, the sum is 7, a prime number. Therefore 7 x 4 = 28 is a perfect number.
If you add 1+2+4+8, the sum 15 is not a prime number, so you can't use Euclid's method here.
If you add 1+2+4+8+16, the sum is 31, a prime number. Therefore 31 x 16 = 496 is another perfect number.
Absolute value of a number:
The absolute value of a number a is | a | and is always positive.
Fibonacci numbers: The Fibonacci numbers is a sequence where
X (n+2) = X (n+1) + X (n), X (1) = 1, X (2) = 1
Example :- 1,1,2,3,5,8,13,21,34,55,89,144..,
it can be clearly seen that any number in the series is the addition of the last two numbers, other than the first two numbers
Example: The price of pens has increased over the years. Each year for the last 7 years the price has increased, and the new price is the sum of the prices for the two previous years. Last year a pen cost 60 rupees. How much does a pen cost today? How much did a pen cost 7 years ago?
Let this year price be x. Last year it was 60, so the previous year it must have been x-60, continuing this process backwards gives us a sequence of expressions:
x, 60, x-60, 120-x, 2x-180, 300-3x, 5x-480, 780-8x, 13x-1260
All of these increases must be positive as every year prize has gone up. That gives us a sequence of inequalities, each of which can be solved to find a range for x:
x > 0
60 > 0
x-60 > 0 x > 60
120-x > 0 x < 120
2x-180 > 0 x > 90
300-3x > 0 x < 100
5x-480 > 0 x > 96
780-8x > 0 x < 97.5
13x-1260 > 0 x > 96.92
Looking at this, we can say
96.92 < x < 97.5
The whole number value x can have is 97, with which we get
x = 97
60 = 60
x-60 = 37
120-x = 23
2x-180 = 14
300-3x = 9
5x-480 = 5
780-8x = 4
13x-1260 = 1
Seven years ago, the price was 4 rupees
In the CAT/MCQ format, where you have the four answers, you can check it by working forward and seeing if the results are correct. You can try putting the given answers for original price and see which one fits in the equation.
Golden ratio: The golden ratio is a special number approximately equal to:
1.6180339887498948482...
Golden ratio = (1 + √ 5)/2
To find the golden ratio, we define the golden ratio as the ratio between x and y if
x y
--- = -----
y x+y
Let's say x is 1. Then we have 1/y = y/(y+1). If we solve this equation to find y, we'll find that it is the value given above, about 1.618
A golden rectangle is a rectangle in which the ratio of the length to the width is the golden ratio.
The concepts like Fibonacci and golden ratio are reference concepts, students are advised not to cram them but just understand the concepts as they are.
Basic Arithmetic Operations:-
Addition, subtraction, multiplication and division are the four basic mathematical operations. We have not gone into details of these concepts as they are very basic; we have added some formulae wherever required. Students preparing for CAT are expected to know the basic arithmetic.
Addition: Addition is used to find the total as a single number of two or more given numbers. The number obtained is called the sum of two numbers.
Subtraction: Subtraction is the quantity left when a smaller number is taken from a greater one. The number obtained is called the difference of two numbers. If a smaller number is subtracted from a greater number, the difference is positive; if a greater numbers is subtracted from a smaller number the result is negative.
Multiplication: Multiplication is the short method of finding the sum of given number of repetitions of the same number. The resultant sum of the repetition is called the product. If one factor is zero then the product is zero. If same factors are multiplied, they can be represented as power or the exponent for example 3 x 3 x 3 = 33
Some short methods in multiplication :
1. multiplication by 11 , 101 , 1001 etc
Rule: add 1, 2, 3 zeroes respectively to the multiplicand and add the multiplicand to the resulting number.
Ex 5023 x 11 = 50230 + 5023 = 55253
i. 5023 x 1001 = 5023000 + 5023 + 5028023
2. Multiplication by 5
Rule: annex a zero to the right of the multiplicand and then divide it by 2
Ex 89356 x 5 = 893560/2 = 446780
3. Multiplication by 25
Rule: annex two zeroes the right of the multiplicand and then divide it by 4
Ex 890023 x 25 = 89002300/4 = 22250575
4. Multiplication by 125
Rule: annex 3 zeroes to the right of the multiplicand and then divide it by 8
5. Multiplication by a number wholly made of nines, i.e. 9 , 99 , 999 etc
Rule: place as many zeroes to the right of the multiplicand as there are nines in the multiplier and from the result subtract the multiplicand.
Ex: 895023 x 999 = 895023000 - 895023 = 894127977.
6. Power Patterns: see the table below and notice the pattern of last digits of powers:
• Pattern of 3: 3, 9, 7, 1 – repeat every four powers
• Pattern of 4: 4,6 – repeat every two powers
• Pattern of 7: 7, 9, 3, 1 – repeat every four powers
• Pattern of 8: 8, 4, 2, 6 – repeat every four powers
• Pattern of 9: 9,1 – repeat every two powers
Application:
Since we have seen the cyclicity of 2,3,7,8 is 4, if we want to find the last digit of any power of these numbers of numbers with last digit as 2,3,7,8 (like 12, 13, 27) can be calculated by finding out remainder of the power divided by four. The last digit of the remainder power will be the last digit of given number.
Examples
Last digit of 232, since 2 has cyclicity of 4, 32/4 has remainder = 0, so the last digit will be same as of 20 or 24, which is 6
Last digit of 325, since 3 has cyclicity of 4, 25/4 has remainder = 1, so the last digit will be same as 31, which is 3
Example: What will be the unit’s digit in 12896?
Ans. 12896 = (12824) (12824) (12824) (12824)
Since we know multiple of 4 of power of 8, last digit is 6
Last digit = 6 × 6 × 6 × 6 = 6
Example: What is the last digit of 22^33^44^55^66^77
Ans. 22^33^44^55^66^77
It can be evaluated by just considering 2 instead of 22 and neglecting higher powers. Any power of 33 × 4n = 3^4n ends in 1 ... that is ... it is of the form 5n + 1 thus 2^(5n + 1) as cyclicity of 2 is 5 .....We will get the last digit as 2 × 1 = 2
Last digit of 66^77 = 6
Last digit of 55^66^77 = 5
Last digit of 44^55^66^77 = 44^(something)25 is same as 44^1 = 4
Last digit of 33^44^55^66^77 = 1
Last digit of 22^33^44^55^66^77 =2
Example: What is the digit in the unit’s place of 251? (CAT 1998)
(a) 2
(b) 8
(c) 1
(d) 4
Ans. (b) The cycle of powers of two is 2,4,8,6 as last digit and repeat. As per that a power of 52(multiple of 4) has last digit of 6, there fore one behind 51 should have last digit of 8.
Division: Division is the method of finding how many times one number called the divisor is contained in another number called dividend. The number of times is called the quotient. The number left after the operation is called the remainder.
(Divisor * quotient) + Remainder = dividend
The number of divisors (including 1 and itself) of a given number N where
N = Am * Bn * Co … where A, B, C are prime numbers are (1+m)(1+n)(1+o)…
Example 2: 90 = 2 * 32 * 5, Here a,b,c are 2,3,5 and m,n,o are 1,2,1. So number of divisors are 2*3*2 = 12, which actually are 1,2,3,5,6,9,10,15,18,30,45,90
Here the sum of the divisors is given by
(a(m+1) – 1)/(a -1) * (b(n+1) – 1)/(b -1) * (c(o+1) – 1)/(c -1) * ….
Taking values from the previous example
(22 – 1)/1 * (33 – 1)/2 * (52 – 1)/4 = 234
Tests for divisibility:
1. A number is divisible by 2 if its unit’s digit is even or zero
2. A number is divisible by 3 if the sum of its digit is divisible by 3.
3. A number is divisible by 4 when the number formed by last two right hand digits is divisible by 4.
4. A number is divisible by 5 if its unit’s digit is 5 or zero
5. A number is divisible by 6 if it’s divisible by 2 and 3 both.
6. Divisibility by 7 has two ways:
Take the last digit, double it, and subtract it from the rest of the number; if the answer is divisible by 7 (including 0), then the number is also. This method uses the fact that 7 divides 2*10 + 1 = 21. Start with the numeral for the number you want to test. Chop off the last digit, double it, and subtract that from the rest of the number. Continue this until you get a one-digit number. The result is 7, 0, or -7, if and only if the original number is a multiple of 7.
Example 3:
123471023473
--> 12347102347 - 2*3 = 12347102341
--> 1234710234 - 2*1 = 1234710232
--> 123471023 - 2*2 = 123471019
--> 12347101 - 2*9 = 12347083
--> 1234708 - 2*3 = 1234702
--> 123470 - 2*2 = 123466
--> 12346 - 2*6 = 12334
--> 1233 - 2*4 = 1225
--> 122 - 2*5 = 112
--> 11 - 2*2 = 7.
This rule holds good for numbers with more than 3 digits is as follows:
Group the numbers in three from unit digit.
add the odd groups and even groups separately
the difference of the odd and even should be divisible by 7
e.g. 85437954 the groups are 85, 437, 954
Sum of odd groups = 954 + 85 = 1039
Sum of even groups = 437
Difference = 602 which is divisible by 7
7. A number is divisible by 8 if the number formed by the last three right hand digits is divisible by 8.
8. A number is divisible by 9 if the sum of its digits is divisible by 9.
9. A number is divisible by 10 if its unit’s digit is zero.
10. To check the divisibility by 11, take the test, alternately add and subtract the digits from left to right. If the result (including 0) is divisible by 11, the number is also. Example: to see whether 365167484 is divisible by 11, start by subtracting: 3-6+5-1+6-7+4-8+4 = 0; therefore 365167484 is divisible by 11
11. A number is divisible by 12 if it’s divisible by 3 and 4 both.
12. A number is divisible by 13 if it fits the following rule:
Delete the last digit from the number, and then subtract 9 times the deleted digit from the remaining number. If what is left is divisible by 13, then so is the original number.
Example: 676, 67 – 6*9 = 13, which is divisible by 13 and so is 676.
13. A number is divisible by 15 when it is divisible by 3 and 5 both. E.g. 930
14. A number is divisible by 25 if the number formed by the last two right hand digits is divisible by 25. e.g. 1025, 3475, 55550 etc.
15. A number is divisible by 125 if the number formed by the last three right hand digits is divisible by 125. e.g. 2125, 4250, 6375 etc.
Example: Which of following numbers are divisible by 12?
(a) 188078 (b) 12496
(c) 3961815 (d) 13685
(e) 28008
Ans. Divisibility rule of 12, number has got to be divisible by 3 and 4
188078, sum of digits = 42, divisible by 3, last two digits not divisible by 4, rejected.
12496, sum of digits = 22, not divisible by 3, rejected
3961815, sum of digits = 33, divisible by 3, last two digits not divisible by 4, rejected.
13685, sum of digits = 23, not divisible by 3, rejected.
28008, sum of digits = 18, divisible by 3, last two digits divisible by 4, it is divisible by 12.
Example: What least number must be added to 127561 so that it is exactly divisible by 28?
Ans. Least number to be added plus the remainder when divided by the given number should give the divisor. Here when we divide 127561 by 28, quotient is4555 and remainder is 21, so 21 + least number = 28, least number = 7
Example: Find the value of ‘a’ and ‘b’ if the seven digit number ‘267a34b’ is divisible by 72.
Ans. For a number to be divisible by 72, it should be divisible by 8 and 9.
Applying rule for 8: number formed by last 3-digits should be divisible by 8.
34b should be divisible by 8, hence
b = 4.
For divisibility by 9: digit sum should by divisible by 9.
Digit sum = 22 + a + b = 26 + a
Hence, a = 1
To be contd...........
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