Page 65 - ICSE Math 8

5 Playing with Numbers

Key Concepts

• Numbers in Generalized Form • Tests of Divisibility
• Reversing the Digits—2-Digit Number • Letters for Digits
• Reversing the Digits—3-Digit Number • Number Puzzles and Games

In this chapter, we will learn about numbers in more detail and deduce the divisibility test rules. We will also
learn how to solve number puzzles.

Numbers in Generalized Form
A number is considered to be in generalized form if it is written as the sum of the product of its digits
with their respective place values. On the other hand, you have already learnt that a natural number can be
expressed in exponential form by utilizing exponents of 10 and the digits of the number. For example, 86
0
1
in exponential form can be written as 8 × 10 + 6 × 10 .
Let’s consider a 2-digit number ab having a and b as the digits at tens and ones Maths Info
places respectively. This can be expressed in the generalized form as 10a + b,
where a (π 0) and b are digits from 0 to 9. Similarly, a 3-digit number abc can The number ab and abc does
be expressed in the generalized form as 100a + 10b + c, where a (π 0), b and not mean a × b or a × b × c.
c are digits from 0 to 9.
Reversing the Digits—2-Digit Number

Let there be a 2-digit number ab, where a (π 0) and b are digits at tens and ones places respectively. On reversing
the digits, we get ba. In the generalized form, ab = (10 × a) + b and ba = (10 × b) + a.
On adding, we get,
ab + ba = (10a + b) + (10b + a)
fi ab + ba = 11a + 11b = 11(a + b) fi ab ba+ =11
ab+
So, it can be deduced that ab + ba is completely divisible by a + b and the quotient is 11. Also, ab + ba is
completely divisible by 11 and the quotient is a + b.
Now, let’s subtract ab and ba (a > b).
ab – ba = (10a + b) – (10b + a)
ab ba−
fi ab – ba = 9a – 9b = 9(a – b) fi = 9
ab−
So, it can be deduced that ab – ba is completely divisible by a – b and the quotient is 9. Also, ab – ba is
completely divisible by 9 and the quotient is a – b.
If, on the other hand, b > a, then find ba – ab.

Example 1: Without actual calculations, write the quotient when the sum of 68 and 86 is divided by
(i) 14 and (ii) 11.
Solution: Here, 68 and 86 are numbers that can be obtained by reversing the digits of the other.
Sum of numbers = 68 + 86 = 154
Sum of digits = 6 + 8 = 14
+
+
Now, 154 ÷ 14 = 11 Ê ab ba ab ba ˆ
ab
and 154 ÷ 11 = 14 Á  ab = 11and 11 =+ ˜
Ë
¯
+
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