Chapter 01 : Number Theory > Topic 2: Basic Arithmatic 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