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Tricks With Digits

Playing With Numbers - Concepts

Numbers in General Form

Tricks With Digits

Interchanging Digits

Divisibility Rule of 2

Divisibility Rule of 10

Divisibility Rule of 5

Divisibility Rule of 9

Divisibility Rule of 3

Divisibility Rule of 8

Divisibility Rule of 6

Divisibility Rule of 11

Divisibility Rule of 4

Cryptarithm

Divisibility Rule of 7

Class - 8th IMO Subjects

Concept Explanation

Tricks With Digits

Trick 1:

Adding the number and its reverse number.

Step 1: Choose a 2 digit number. Let us take 34

Step 2: Now reverse the digits. We get 43

Step 3: Add the two digits . we get 34 + 43 = 77

Step 4 : Divide the sum by 11. We get 77 $div$ 11 = 7

Step 5 : The quotient will always be equal to the sum of digits that is 3 and 4 in this case

Justification:

Consider a two digit number $overline{ab}$ having ones and tens digits as b and a respectively. On reversing the digits of this number , we obtain a two digit number $overline{ba}$ .

In expanded form, we have

$overline{ab}= 10a+b$

and, $overline{ba}= 10b+a$

Adding (i) and (ii) , we get

$overline{ab}+overline{ba}=(10a+b)+(10b+a)$

$Rightarrow ;;overline{ab}+overline{ba}=(11a)+(11b)$

$Rightarrow ;;overline{ab}+overline{ba}=11(a+b)$

$Rightarrow ;;frac{overline{ab}+overline{ba}}{11}=a+b;;and;;frac{overline{ab}+overline{ba}}{a+b}=11$

Thus, $overline{ab}+overline{ba}$ is completely divisible by 11 and the quotient is a+b.

Also, it is divisible by a+b and in that case the quotient is 11.

In other words, the sum of any two digit number $overline{ab}$ and the number $overline{ba}$ by reversing its digits is completely divisible by

(i) the sum a+b of its digits and the quotient is 11.

(ii) 11 and the quotient is a+b i.e., the sum of its digits.

Trick 2:

Subtracting the number and its reverse number.

Step 1: Choose a 2 digit number. Let us take 28

Step 2: Now reverse the digits. We get 82

Step 3: Subtract the smaller number from the bigger number . We get 82 - 28= 54

Step 4 : Divide the difference by 9. We get 54 $div$ 9 = 6

Step 5 : The quotient will always be equal to the difference of the greater digit and the smaller digit that is 8 and 2 in this case. 8 -2 = 6

Justification:

Consider a two digit number $overline{ab}$ having ones and tens digits as b and a respectively. On reversing the digits of this number , we obtain a two digit number $overline{ba}$ .

In expanded form, we have

$overline{ab}= 10a+b$

and, $overline{ba}= 10b+a$

Subtracting (ii) from (i) , we get

$overline{ba}-overline{ba}=(10a+b)-(10b+a)$

$Rightarrow ;;overline{ab}-overline{ba}=(9a)-(9b)$

$Rightarrow ;;overline{ab}-overline{ba}=9(a-b)$

$Rightarrow ;;frac{overline{ab}-overline{ba}}{9}=a-b;and;frac{overline{ab}-overline{ba}}{a-b}=9$

Thus, $overline{ab}-overline{ba}$ is exactly divisible by 9 and the quotient is a- b i.e. the difference of the digits.

Also, $overline{ab}-overline{ba}$ is exactly divisible by a- b (difference of digits) and the quotient is 9.

Illustration 1: Without performing actual addition and division find the quotient when the sum of 79 and 97 is divided by (i) 16 (ii) 11.

Solution: The two numbers 79 and 97 are such that one can be obtained reversing the digits of the other.

Therefore, their sum when divided by the sum of the digits i.e. 7 + 9 = 16, we obtain 11 as the quotient .

If the sum of these two numbers is divided by 11, we get 7 + 9 ( sum of the digits) =16 as the quotient.

Illustration 2 : Without performing actual calculation find the quotient when the difference of 73 and 37 is divided by (i) 4 (ii) 9.

Solution: The two numbers 73 and 37 are such that one can be obtained reversing the digits of the other.

Therefore, their difference 73 - 37 = 36 when divided by the difference of the digits i.e. 7 - 3 = 4, we obtain 9 as the quotient .

If the difference 73 - 37 = 36 is divided by 9, we get 7 - 3 ( difference of the digits) = 4 as the quotient.

Sample Questions

(More Questions for each concept available in Login)

Question : 1

Find the product of the quotient when the sum of 28 and 82 is divided by 10 and the difference of 82 and 28 is divided by 6.
A. 99	B. 9	C. 11	D. 7
Right Option : A
View Explanation

Question : 2

Without performing actual addition and division find the quotient when the sum of 68 and 86 is divided by 11
A. 11	B. 9	C. 14	D. 7
Right Option : C
View Explanation

Question : 3

Without performing actual calculation find the quotient when the difference of 71 and 17 is divided by 6.
A. 11	B. 9	C. 12	D. 7
Right Option : B
View Explanation