Page 40 - Start Up Mathematics_6

Example 12: If the product of two whole numbers is one, can we say that one or both of them
will be one? Justify. (NCERT)

Solution: (a) Let’s assume that the two whole numbers ‘a’ and ‘b’ are such that a = 1 and
b ≠ 1. Now a × b = 1 × b = b ≠ 1. Hence a = 1 and b ≠ 1 is not possible.

(b) Let’s take a = 1 and b = 1.
Now, a × b = 1 × 1 = 1.

Since no other values of a and b give the product 1, hence the only possibility
is that both the whole numbers a and b should be 1.

IV. Division

 Let ‘a’ and ‘b’ be any two whole numbers such that b ≠ 0, then a ÷ b is not necessarily a whole
number.
Division of whole numbers is not closed.

For example, 9 ÷ 3 = 3, which is a whole number
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22 ÷ 5 = , which is not a whole number
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 Let ‘a’ and ‘b’ be non-zero whole numbers, then a ÷ b ≠ b ÷ a.
Division of whole numbers is not commutative.

For example, 8 ÷ 2 = 4, but 2 ÷ 8 is not a whole number

 Let ‘a’, ‘b’ and ‘c’ be any three whole numbers, then (a ÷ b) ÷ c ≠ a ÷ (b ÷ c).
Division of whole numbers is not associative.

For example, (24 ÷ 6) ÷ 2 ≠ 24 ÷ (6 ÷ 2)

LHS = (24 ÷ 6) ÷ 2 = 4 ÷ 2 = 2; RHS = 24 ÷ (6 ÷ 2) = 24 ÷ 3 = 8
 Let ‘a’ be any whole number, then a ÷ 1 = a.

Any whole number ‘a’ when divided by 1 gives the quotient ‘a’.
For example, 6 ÷ 1 = 6

 Let ‘a’ be any non-zero number, then a ÷ a = 1.

Any non-zero whole number ‘a’ when divided by itself gives the quotient 1.
For example, 15 ÷ 15 = 1

 When zero is divided by any non-zero whole number, the quotient is always zero.

For example, 0 ÷ 9 = 0
 Let ‘a’, ‘b’ and ‘c’ be any three whole numbers such that b ≠ 0 and a ÷ b = c, then b × c = a.

For example, 16 ÷ 8 = 2, then 8 × 2 = 16

 Let ‘a’, ‘b’ and ‘c’ be any three whole numbers such that b ≠ 0, c ≠ 0 and b × c = a, then
a ÷ b = c and a ÷ c = b.
For example, 2 × 3 = 6, then 6 ÷ 2 = 3 and 6 ÷ 3 = 2

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