Introduction
Numbers have various properties that help in performing operations efficiently. In this chapter, we will explore divisibility, factors, multiples, and different tests of divisibility.
3.1 Factors and Multiples
Factor: A number that divides another number completely without leaving a remainder.
Multiple: The result of multiplying a number by another whole number.
Example:
Factors of 12: 1, 2, 3, 4, 6, 12
Multiples of 4: 4, 8, 12, 16, 20, …
| Number | Factors | Multiples |
|---|---|---|
| 6 | 1, 2, 3, 6 | 6, 12, 18, 24 |
| 8 | 1, 2, 4, 8 | 8, 16, 24, 32 |
3.2 Prime and Composite Numbers
Prime Number: A number that has exactly two factors (1 and itself). Example: 2, 3, 5, 7, 11
Composite Number: A number that has more than two factors. Example: 4, 6, 8, 9, 10
Special Case: 1 is neither prime nor composite.
3.3 Tests of Divisibility
Divisibility Rules:
By 2: A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
By 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
By 5: A number is divisible by 5 if it ends in 0 or 5.
By 10: A number is divisible by 10 if it ends in 0.
By 11: A number is divisible by 11 if the difference between the sum of digits at odd and even places is 0 or a multiple of 11.
Example:
Is 432 divisible by 3?
Sum of digits: 4 + 3 + 2 = 9 (which is divisible by 3)
So, 432 is divisible by 3.
3.4 Common Factors and Common Multiples
Common Factors: Factors that two or more numbers share.
Common Multiples: Multiples that two or more numbers share.
Example:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common Factors: 1, 2, 3, 6
Multiples of 4: 4, 8, 12, 16, 20, 24, …
Multiples of 6: 6, 12, 18, 24, 30, …
Common Multiples: 12, 24, 36…
3.5 Prime Factorization
Breaking down a number into prime factors.
Example: 36 = 2 × 2 × 3 × 3
3.6 Highest Common Factor (HCF) & Lowest Common Multiple (LCM)
HCF: The largest factor common to two or more numbers.
LCM: The smallest multiple common to two or more numbers.
Finding HCF using Prime Factorization:
Example: HCF of 18 and 24
Prime factorization:
18 = 2 × 3 × 3
24 = 2 × 2 × 2 × 3
Common factors: 2, 3
HCF = 2 × 3 = 6
Finding LCM using Prime Factorization:
Example: LCM of 18 and 24
Prime factorization:
18 = 2 × 3 × 3
24 = 2 × 2 × 2 × 3
LCM = 2 × 2 × 2 × 3 × 3 = 72
3.7 Applications of HCF and LCM
HCF is used for: Dividing things into equal parts.
LCM is used for: Finding the least number where multiple events occur together.
Example:
Two lights blink every 15 and 20 seconds. When will they blink together?
LCM of 15 and 20 = 60 seconds.
Conclusion
Understanding factors, multiples, and divisibility helps in solving real-life problems and arithmetic operations efficiently.