# Playing with Numbers-Notes

Class VIII Math
Notes for Playing with Numbers
6.1   Numbers in general form
A 2 digit number can always be written as a combination of 2 different numbers.
for eg:–
65 = 10 × 6 + 5 → 6 is at tens place and 5 is at ones place.
23 = 10 × 2 + 3 → 2 is at tens place and 3 is at ones place
Thus, any two digit number can be written in a general form as 10 × x + y
Similarly,
572 = 5 × 100 + 7 × 10 + 2
123 = 1 × 100 + 2 × 10 + 3
The three digit number xyz can be written as 100 × x + 10 × y + z
6.2   Puzzles & games
Puzzles and games are a source of entertainment and education that makes interesting and challenging situations.
Reversing the digits of a two-digit number
Step-1 : Choose any 2-digit number of the form 10 x + y.
Step-2 : Reverse the digits to get a new number i.e.,
Step-3: Add the reversed number to the original number.
(10x + y) + (10y +x) = 11x + 11y = 11(x +y)
Step-4 : Divide the answer by 11.
11(x + y) ÷ 11 = (x + y)
Result: There is no remainder.
Remark: The sum of a two-digit number and the number formed by reversing its digits is exactly divisible by 11 and the quotient obtained is the sum of the digits of the original 2-digit number.
Adding both the number, we get 36 + 63 = 99, which is exactly divisible by 11
∴ Hence proved.
Subtraction
Step-1 : Choose a two digit number in the form 10x + y.
Step-2 : Reverse the digits to get a new number in the form 10y + x.
Step-3 : Subtract both the numbers.
(10y + x) – (10x + y) = 9y – 9x = (9 (y – x)
Step-4 : Divide the answer by 9.
9(y – x) ÷ 9 = (y – x)
Result: There is no remainder.
Remark: The difference of a two digit number and its reversed number is exactly divisible by 9 and the quotient obtained is either the difference of the digits of the original 2-digit number or 0.
Reversing the digits of a three-digit number
Step-1 : Choose any three-digit number xyz in the form 100x + 10y + z
Step-2 : From 2 more numbers in a way
yzx = 100z + 10x + y.
Step-3 : Add all the three numbers
(100x + 10y + z) + (100y + 10z + x)
+ (100z + 10x + y)
Step-4 : Divide the answer by 111.
= 111 (x + y + z) ÷ 111 = (x + y + z).
Result: There is no remainder.
Remark: The sum of a 3-digit number and the number formed by arranging its digits in such a way that each digit occupies a place value only once, is exactly divisible by 111 and the quotient obtained is the sum of the digits of the original 3-digit number.
Step-1 : Take any three-digit number xyz in the form 100x + 10y + z.
Step-2 : Reverse the digits : zyx = 100z + 10y + x.
Step-3 : Substract both the numbers.
(100x + 10y + z) – (100z + 10y + x)
= 99x – 99z = 99(x – z)
Result: There is no remainder.
Remark: The difference of a 3-digit number and the number formed by reversing the digits is exactly divisible by 99 and the quotient so obtained is either the difference between the hundredth digit and the ones digit of the original 3-digit number or 0.
6.3   Letter for digits
Every game has same rules. So, there are some rules for such puzzles also. There are two rules for solving them.
(i) Each letter in the puzzle must stand for just one digit. Each digit must be represented by just one letter.
(ii) The first digit of a number cannot be zero.
Numerical Ability 6.1
Solve for Q:

Solution:
From the addition above, we can say Q + 3 = 1. For this, Q must be equal to 8. So, the puzzle becomes:

Numerical Ability 6.2
Find the digits A, B and C.

Solution:
Since the one�s digit of B × 3 is B, it must be B = 0 or B = 5.
Now, for A
If A = 1

These two are not possible because C cannot be zero.
If A = 2

These two are not possible because C cannot be zero.
If A = 3

If is not possible because A is not zero and C cannot be zero.
If A = 5

6.4   Tests of divisibility
Divisible by 2
A number is divisible by 2, if its unit digit is 0 or divisible by 2 i.e., 2, 4, 6, 8
DIY.
Find the condition when a two-digit number xy and a 3-digit number xyz will be exactly divisibly by 2.
Explanation
2-digit number xy can be written as 10x + y. 2 will always divide 10x.
So, 10x + y will be exactly divisible by 2 if y = 0, 2, 4, 6 or 8.
A 3-digit number xyz can be written as 100x + 10y + z. We can say, 2 will always divide 100x and 10y. So,
100x + 10y + z will be divisible by 2 if z = 0, 2, 4, 6 or 8.
Divisible by 3
A number is divisible by 3, if the sum of its digits is divisible by 3.
Divisible by 5
A number is divisible by 5, if the digit in its unit�s place is 5 or zero.
Divisible by 9
A number is divisible by 9, if the sum of its digits is divisible by 9.
Divisible by 10
If number is divisible by 10, if the digit at unit’s place is zero.