(v) 3, 6, 15, 27. Here 3 is the only number that divides them all and
is the HCF in this case.
C. (i) Since 3, 7 are both primes, the LCM is simply their product, 21.
(ii) 9= 3 ×3, so 3 and 9 both divide 9 and there is no smaller number
that does so. The LCM is thus 9.
(iii) Since 3× 12 =36 and 2× 18 =36, we see directly that the LCM
is 36.
(iv) We have to deal with 3, 5, and 9= 32. The LCM must contain at
least two factors of 3 and one of 5. So the LCM is 5× 32 =45.
(v) 2, 4, 6. 2 divides 4, so we must have a factor of 4 in the LCM.
Also, 4 and 6 both divide 12, but no smaller number and so the
LCM is 12.1.2.4 Manipulation of numbers
➤
328 ➤Much of arithmetic is based on just a few operations: addition, subtraction, multiplication
and division, satisfying a small number of rules. The extension of these rules to include
symbolsas well as numbers leads us on toalgebra(Chapter 2).
Addition, denoted+, produces thesumof two numbers:
6 + 3 = 9 = 3 + 6 (addition is ‘commutative’)Subtraction, denoted−, produces thedifferenceof two numbers:6 − 3 = 3 =−( 3 − 6 ) (minus sign changes signs in brackets)Multiplication, denoted bya×bor simply asabin algebra, produces theproductof
two numbers:
6 × 3 =( 6 )( 3 )= 18 = 3 × 6 (multiplication is commutative)a·bis sometimes used to denote the product but can be confused with decimal notation
in arithmetic.
Division, denoted bya÷bora/bor, better,
a
b, produces thequotientof two numbers:6 ÷ 3 = 6 / 3 =^63 = 2 (of course, 6 ÷ 3 = 3 ÷6!)Note that÷and / are very rarely used in written calculations, where we use the form
6
3 unless we need to call into play÷or / because we have a large number of divisions.
Also notice how we have simplified the quotient to 2. We always simplify such fractions
to lowest form whenever we can (➤12).
The way the above and other arithmetic operations are combined is according to a set
of conventional precedences –the rules of arithmetic. Thus we always perform multi-
plication before addition, so:
2 × 3 + 5 = 6 + 5 = 11