Factors, Multiples, LCM-HCF
Factors, Multiples, LCM-HCF
A factor divides a number exactly with no remainder; a multiple is what you get by multiplying a number by a whole number. HCF is the largest number that divides two numbers exactly, and LCM is the smallest number that both divide into exactly.
Factors and multiples
- Factors of 12: 1, 2, 3, 4, 6, 12. Every number has 1 and itself as factors.
- Multiples of 12: 12, 24, 36, 48 and so on, endlessly.
- A number has a limited number of factors but an unlimited number of multiples.
- The smallest factor of any number is 1; the smallest multiple of a number is the number itself.
- Prime numbers have exactly two factors, 1 and themselves. 2 is the smallest prime and the only even prime. 1 is neither prime nor composite.
HCF and LCM
| Point | HCF | LCM |
|---|---|---|
| Full form | Highest Common Factor | Lowest Common Multiple |
| What it is | Largest number dividing both exactly | Smallest number divisible by both |
| Size | Not greater than the smaller number | Not smaller than the larger number |
| Used for | Cutting into largest equal pieces, largest equal groups | Bells ringing together, meeting again, common denominator |
HCF x LCM = product of the two numbers
For 12 and 18: HCF is 6, LCM is 36, and 6 x 36 = 216 = 12 x 18. This relation holds for two numbers only, not for three.
Finding them by prime factorisation
12 = 2 x 2 x 3 and 18 = 2 x 3 x 3. For the HCF take the common factors with the lowest power: 2 x 3 = 6. For the LCM take all factors with the highest power: 2 x 2 x 3 x 3 = 36.
Reading the word problem correctly
- 'Largest' or 'maximum' size, 'greatest number of children who can each get an equal share' — this is HCF.
- 'Least' or 'minimum' number, 'when will they ring together again', 'smallest number of sweets that can be divided equally among 6 or 8' — this is LCM.
✗ The HCF of two numbers is always larger than both. | ✓ The HCF is never larger than the smaller number; the LCM is never smaller than the larger.
संकेत: महत्तम समापवर्तक (HCF) = सबसे बड़ी बाँटने वाली संख्या; लघुत्तम समापवर्त्य (LCM) = सबसे छोटी साझा गुणज संख्या।
Expect 1-2 questions, usually a word problem that hides which of HCF and LCM is wanted. Read for the words largest and smallest — largest equal pieces means HCF, next time together means LCM. The relation HCF x LCM equals the product is asked directly and is also a fast way to find one when the other is known. The standard trap gives three numbers with that relation, where it does not hold. Another trap calls 1 a prime number.
Frequently asked questions
What is the difference between HCF and LCM?
The HCF is the largest number that divides both given numbers exactly, while the LCM is the smallest number that both given numbers divide exactly. HCF is never bigger than the smaller number and LCM is never smaller than the larger.
Is 1 a prime number?
No. A prime number has exactly two distinct factors, one and itself. The number 1 has only one factor, so it is neither prime nor composite. The smallest prime number is 2, which is also the only even prime.
When should a word problem be solved by LCM?
When it asks for the least or smallest number that satisfies a shared condition — bells ringing together again, the smallest quantity that can be divided equally among different groups, or when two cyclists next meet at the starting point.
60-second recap
- A factor divides exactly; a multiple is a product with a whole number.
- Factors are finite; multiples are infinite. 1 is neither prime nor composite.
- HCF = largest common factor; LCM = smallest common multiple.
- HCF x LCM = product of the two numbers, for two numbers only.
- Largest or maximum means HCF; least or together again means LCM.