Chapter 3: Numbers, Factors, Multiples & Prime Factorization
संख्याएं, गुणनखंड, गुणज, ल.स.प. (LCM) एवं म.स.प. (HCF)
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- Prime vs Composite: Prime numbers have exactly 2 distinct factors (1 and itself). 1 is neither prime nor composite. 2 is the ONLY even prime number.
- Twin Primes & Co-Primes: Twin primes are prime numbers differing by 2 (e.g. 3 & 5, 5 & 7, 11 & 13, 17 & 19). Co-prime numbers are pairs of numbers having NO common factor other than 1 (e.g. 8 and 15 are co-prime even though neither is prime).
- LCM (Least Common Multiple) Word Problems: Traffic signal lights blinking at intervals of 12s, 15s, and 20s will blink together again at their LCM (60 seconds).
- HCF (Highest Common Factor) Word Problems: Finding the maximum length of a measuring tape or largest square tile to pave a rectangular courtyard without cutting requires calculating HCF.
- Divisibility Rules: Divisibility by 3 and 9 (sum of digits); Divisibility by 4 and 8 (last 2 or 3 digits); Divisibility by 11 (difference between sum of odd-placed digits and even-placed digits is 0 or multiple of 11).
- Trap: Is 1 a prime number? NO. 1 is neither prime nor composite because it has only one factor.
- Trap: Are co-prime numbers always prime? NO. 4 and 9 are co-prime because HCF(4, 9) = 1, but both 4 and 9 are composite.
- Trap: In periodic simultaneous event word problems (bells, lights, runners), ALWAYS find **LCM**.
NCERT Core Facts: Factors, Multiples & Arithmetic Algorithms
In CTET Paper 1 and Paper 2, number theory word problems test practical arithmetic logic:
1. Prime Number Foundations
- Smallest Prime Number: 2 (the only even prime).
- Primes from 1 to 100: Exactly 25 prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97).
- Twin Primes: Prime pairs with a difference of 2: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).
2. Deciding Between LCM and HCF in Word Problems
| Problem Scenario Clue | Mathematical Strategy | Formula / Example |
|---|---|---|
| Simultaneous Events: "Three bells toll at intervals of 9, 12, and 15 minutes. When will they toll together next?" | LCM (Least Common Multiple) | Find \text{LCM}(9, 12, 15) = 180\text{ mins} = 3\text{ hours}. |
| Maximum Capacity / Division: "Find the greatest measure of a container that can measure milk of three cans (403L, 434L, 465L) exactly." | HCF (Highest Common Factor) | Find \text{HCF}(403, 434, 465) = 31\text{ Liters}. |
| Product Relationship: | \text{LCM}(a, b) × \text{HCF}(a, b) = a × b | Product of two numbers equals product of LCM and HCF. |
Test Your Mastery: Numbers, Factors, Multiples & Prime Factorization
Authentic CBSE exam questions tested directly from this NCERT chapter.
Three traffic signal lights change after every 48 seconds, 72 seconds, and 108 seconds respectively. If they all change simultaneously at 8:20:00 hours, at what time will they change together next?
तीन ट्रैफिक बत्तियाँ क्रमशः प्रत्येक 48 सेकंड, 72 सेकंड और 108 सेकंड के बाद बदलती हैं। यदि वे 8:20:00 बजे एक साथ बदलती हैं, तो अगली बार वे किस समय एक साथ बदलेंगी?
Which of the following statements concerning prime and composite numbers is TRUE?
अभाज्य और भाज्य संख्याओं के संदर्भ में निम्नलिखित में से कौन-सा कथन सत्य है?
How many prime numbers exist between 1 and 50?
There are exactly 15 prime numbers between 1 and 50.
When should LCM be applied in word problems?
Whenever events repeat periodically and you need to find when they will coincide or occur together simultaneously.