Numbers have fascinating properties related to factors, multiples, divisibility, prime and composite numbers, and common factors/multiples. This chapter explores these ideas.
Factors and Multiples
A factor of a number divides it exactly (no remainder). For example, factors of 12 are 1, 2, 3, 4, 6, 12.
A multiple of a number is obtained by multiplying it by a natural number. Multiples of 5: 5, 10, 15, 20, ...
Prime and Composite Numbers
A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.
A composite number has more than two factors. Examples: 4, 6, 8, 9, 10.
Note: 1 is neither prime nor composite. 2 is the only even prime number.
- Divisibility Rules
- By 2: last digit is 0, 2, 4, 6, or 8.
- By 3: sum of digits is divisible by 3.
- By 4: last two digits form a number divisible by 4.
- By 5: last digit is 0 or 5.
- By 6: divisible by both 2 and 3.
- By 8: last three digits divisible by 8.
- By 9: sum of digits divisible by 9.
- By 10: last digit is 0.
- By 11: difference of sums of alternate digits is 0 or divisible by 11.
Find all factors of 36.
36 = 1 x 36 = 2 x 18 = 3 x 12 = 4 x 9 = 6 x 6.
Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Is 561 divisible by 3?
Sum of digits = 5 + 6 + 1 = 12. 12 is divisible by 3, so yes, 561 is divisible by 3.
HCF (Highest Common Factor)
The HCF of two or more numbers is the greatest factor common to all of them.
Method: List all factors of each number and find the greatest common one.
Find HCF of 24 and 36.
Factors of 24: 1,2,3,4,6,8,12,24. Factors of 36: 1,2,3,4,6,9,12,18,36.
Common factors: 1,2,3,4,6,12. HCF = 12.
LCM (Lowest Common Multiple)
The LCM of two or more numbers is the smallest multiple common to all of them.
Find LCM of 4, 6, and 12.
Multiples of 4: 4,8,12,16,... Multiples of 6: 6,12,18,... Multiples of 12: 12,24,...
LCM = 12.
Find LCM and HCF of 15 and 20 by prime factorisation.
15 = 3 x 5; 20 = 22 x 5.
HCF = 5 (common factors). LCM = 22 x 3 x 5 = 60.
Check: Is 74,132 divisible by 4?
Last two digits = 32. 32 / 4 = 8. Yes, divisible by 4.
Two bells ring at intervals of 8 and 12 minutes. If they ring together at 9:00 a.m., when will they next ring together?
LCM of 8 and 12 = 24 minutes. They will ring together at 9:24 a.m.
Key Formula
For any two numbers: HCF x LCM = Product of the two numbers.
Common mistakes
- 1 is not a prime number.
- Every number is a factor of itself.
- HCF is always less than or equal to the smaller number; LCM is always greater than or equal to the larger number.
Summary
Factors divide a number exactly; multiples are results of multiplication. Prime numbers have exactly two factors. Divisibility rules offer quick tests. HCF is the greatest common factor; LCM is the smallest common multiple.