Playing With Numbers Class 8 Notes - Chapter 16

General Form of Numbers

Numbers follow interesting properties, and today we will learn about a few of them.

Let us understand how numbers can be written in the general form.

The expression which shows a number as the sum of the products of its digits and their place values is the general form of the number.

General Form

A two-digit number ‘ab’ can be written in the form: ab = 10a + b.
A three-digit number ‘abc’ can be written in the form: abc = 100a + 10b + c.

Example:

268 = 200 + 60 + 8
= 100 × 2 + 10 × 6 + 8

Now, let us see some interesting properties shown by the numbers.

Properties

Reversing the digits of a two-digit number and adding

If we reverse the digits of a two-digit number and add the new number with the original number, the sum obtained will be perfectly divisible by 11. That is, if we divide the sum by 11, the remainder will be 0. The quotient will be the sum of the digits of the number.

Reversing the digits of a two-digit number and subtracting

If we reverse the digits of a two-digit number and subtract the smaller number from the larger number, the difference obtained will be perfectly divisible by 9. In other words, if we divide the difference by 9, the remainder will be 0. The quotient will be the difference between the digits of the number (smaller digit subtracted from the larger digit).

Reversing the digits of a three-digit number

If we reverse the digits of a three-digit number and subtract the smaller number from the larger number, the difference obtained will be perfectly divisible by 99. In other words, if we divide the difference by 99, the remainder will be 0. The quotient will be the difference between the first and the third digit of the original number (smaller number subtracted from the larger number).

NCERT Solutions for Class 8 Maths Chapter 16
NCERT Solutions Class 8 Maths Chapter 16
NCERT Solutions Class 8 Maths Exercise 16.1
NCERT Solutions Class 8 Maths Exercise 16.2

Forming three-digit numbers with the given three digits

If we take some particular combinations of a three-digit number, such that all the digits are different, and then we add the combination of the digits, the resulting number will be perfectly divisible by 37. In other words, if we divide the sum by 37, the remainder will be 0. If you choose the number abc, then the sum of abc, cab and bca will always be divisible by 37.

We can also form interesting puzzles with the help of numbers. Let us see how we can do this.

Letters for Digits

Now we will solve puzzles. Here, the letters will represent the digits, and we have to find which digit is represented by each letter.

Rules to solve the puzzles:

• Each letter can represent only one digit.
• The first digit of the number cannot be zero.
• The puzzle will have just one answer.

Example

Find the value of P and Q in the sum given below.
6p + 4Q = 78
Here, the sum of the unit digits, P and 4 is 8.

So, P = 4.

For the next step, the sum of Q and 6 is 7.

So, Q = 1.

Hence, the value of P is 4 and Q is 1.

Now, let us read about divisibility tests.

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Divisibility Tests

We can test the divisibility of some numbers based on the unit's digit of the given number. Whereas, for some numbers, it is based on the sum of all the digits of the given number. Let us see the different rules.

Divisibility by 10: A number is divisible by 10 if the unit digit of the number is zero.
Divisibility by 5: A number is divisible by 5 if the unit digit of the number is zero or five.
Divisibility by 2: A number is divisible by 2 if the unit digit of the number is divisible by 2 or the number is even.
Divisibility by 3: A number is divisible by 3 only if the sum of the digits of the number is divisible by 3.
Divisibility by 9: A number is divisible by 9 only if the sum of the digits of the number is divisible by 9.
Divisibility by 4: If the number formed by the last two digits of the given number, in that order, is a multiple of 4, then the whole number is divisible by 4.
Divisibility by 8: If the number formed by the last three digits of the given number, in that order, is a multiple of 8, then the whole number is divisible by 8.
Divisibility by 6: If a number is divisible by 2 and 3 both, then the number is also divisible by 6.
Divisibility by 11: If the difference of the sum of the digits at odd place and the sum of the digits at even place of the number is 0 or a multiple of 11, then the number is divisible by 11.

Let us see some examples.

Is 2717 divisible by 11?.

The sum of the alternate digits starting from the left will be:
2 + 1 = 3
7 + 7 = 14

Difference is 14 – 4=3 = 11

Since, the difference is 11, therefore 2717 is divisible by 11.

Is 23716 divisible by 4?

Since, the number formed by the last two digits is 16, which is divisible by 4; therefore, 23716 is also divisible by 4.

Is 93216 divisible by 3 and 9?

Sum of the digits will be:

9 + 3 + 2 + 1 + 6 = 21

Since, 21 is divisible by 3 but not by 9, therefore, 93216 is divisible by 3 but not by 9.

Links to Other Resources for Class 8 Maths

Other Class 8 Maths Related Links
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