Page 37 - ICSE Math 6

Example 9: If the product of two whole numbers is one, can we say that one or both of them will
be one? Justify.

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 are 1.

IV. Division
• Closure property
Let ‘a’ and ‘b’ be any two whole numbers such that b ≠ 0, then a ÷ b is not necessarily a whole
number.
Thus, whole numbers are not closed under division.
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For example, 22 ÷ 5 = , which is not a whole number.
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• Commutative property
Let ‘a’ and ‘b’ be non-zero whole numbers, then a ÷ b ≠ b ÷ a.

Thus, whole numbers are not commutative under division.
For example, 8 ÷ 2 = 4, but 2 ÷ 8 is not a whole number.
• Associative property
Let ‘a’, ‘b’ and ‘c’ be three whole numbers where b ≠ 0 and c ≠ 0, then (a ÷ b) ÷ c ≠ a ÷ (b ÷ c).

Thus, whole numbers are not associative under division.
For example, (24 ÷ 6) ÷ 2 ≠ 24 ÷ (6 ÷ 2) Maths Info
LHS = (24 ÷ 6) ÷ 2 = 4 ÷ 2 = 2; RHS = 24 ÷ (6 ÷ 2) = 24 ÷ 3 = 8 Under division, the identity and
• Any whole number ‘a’ when divided by 1 gives the quotient ‘a’. inverse of whole numbers do

For example, 6 ÷ 1 = 6 not exist.
• Any non-zero whole number 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

Division Algorithm y q
x
If a whole number x is divided by a non-zero whole number y, then there exists a quotient –qy
q and a remainder r such that x = yq + r, where q and r are whole numbers and r < y. r

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