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Class 6 Mathematics Chapter 5 Prime Time — Formulas & Key Points

Chapter 5 Prime Time in NCERT Class 6 Mathematics introduces the building blocks of number theory: factors, multiples, primes, composites, divisibility rules, HCF and LCM. These concepts underpin mental arithmetic, fraction simplification and algebra in higher classes. This formula sheet presents every definition, test and method in table format with memory aids and worked examples to speed up revision.

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Key takeaways

  • ✓Every whole number greater than 1 is either prime (exactly two factors: 1 and itself) or composite (more than two factors).
  • ✓Divisibility tests for 2, 3, 5, 9, 10 and 11 let you quickly check if a number is divisible without performing full division.
  • ✓Prime factorisation expresses any composite number as a product of prime numbers; use factor trees or repeated division.
  • ✓HCF (Highest Common Factor) is the largest number that divides two or more numbers; find it via prime factorisation (product of lowest powers) or continued division.
  • ✓LCM (Lowest Common Multiple) is the smallest number divisible by two or more numbers; calculate using prime factorisation (product of highest powers) or the formula HCF × LCM = Product of two numbers.
  • ✓The Sieve of Eratosthenes is an ancient algorithm to list all primes up to a given limit by systematically eliminating multiples.
  • ✓Co-prime (or relatively prime) numbers have HCF equal to 1; they need not be prime themselves.

Core Definitions and Terminology

Understanding precise terminology is the foundation of Prime Time. A factor of a number divides it exactly without leaving a remainder; for example, 1, 2, 3, 4, 6 and 12 are all factors of 12. A multiple is obtained by multiplying a number by any whole number; multiples of 5 are 5, 10, 15, 20 and so on. A prime number has exactly two distinct factors: 1 and the number itself. Examples include 2, 3, 5, 7, 11, 13. A composite number has more than two factors; 4, 6, 8, 9, 10, 12 are composites. The number 1 is special—it is neither prime nor composite. Two numbers are co-prime if their only common factor is 1; for instance, 8 and 15 are co-prime even though neither is prime. Every even number except 2 is composite, and 2 is the only even prime. These definitions recur throughout CBSE Class 6 Mathematics Chapter 5 and must be memorised verbatim for objective questions.
  • Factor — a number that divides another number exactly (remainder zero)
  • Multiple — the product of a number and any whole number
  • Prime number — has exactly two factors: 1 and itself (2, 3, 5, 7, 11…)
  • Composite number — has more than two factors (4, 6, 8, 9, 10…)
  • Co-prime numbers — HCF is 1; they share no common prime factor
  • Unit — the number 1 is neither prime nor composite

Divisibility Tests — Complete Table

Divisibility rules help you quickly check whether one number divides another without performing long division—a vital skill for simplifying fractions and finding HCF and LCM. The NCERT Class 6 Mathematics syllabus covers six tests: for 2, 3, 5, 9, 10 and 11. Each rule examines specific digit positions or sums. Mastery of these tests saves time in mental maths and board exams. For instance, the test for 3 and 9 both require adding all digits; for 3 the sum must be divisible by 3, for 9 the sum must be divisible by 9. The test for 11 is unique: subtract the sum of digits in odd places from the sum in even places; if the result is 0 or divisible by 11, the original number is divisible by 11. Always apply these tests before attempting prime factorisation to simplify your work and avoid calculation errors.
  • Test for 2 — last digit is 0, 2, 4, 6 or 8 (any even digit)
  • Test for 3 — sum of all digits is divisible by 3
  • Test for 5 — last digit is 0 or 5
  • Test for 9 — sum of all digits is divisible by 9
  • Test for 10 — last digit is 0
  • Test for 11 — (sum of digits at odd places) − (sum of digits at even places) is 0 or divisible by 11

Prime Factorisation Methods

Prime factorisation expresses any composite number as a product of prime numbers. Two standard methods appear in CBSE Class 6 Mathematics notes: the factor tree and the division method. In a factor tree, write the number at the top, split it into any two factors, then continue splitting each branch until all end-nodes are prime. Circle or underline the primes. In the division method, divide the number by the smallest prime (2, then 3, then 5…) repeatedly until the quotient is 1; write the divisors in a column and multiply them. Both yield the same prime factorisation. Prime factorisation is the gateway to finding HCF and LCM efficiently. Always write the final answer in exponential form when primes repeat; for example, 72 equals two raised to three times three squared, written as 2³×3². This compact notation prevents mistakes in further calculations.
  • Factor tree — split the number into branches until all leaves are prime; multiply the primes
  • Division method — divide by smallest primes (2,3,5,7…) successively until quotient is 1
  • Express answer in exponential form: 2³×3² instead of 2×2×2×3×3
  • Every composite has a unique prime factorisation (Fundamental Theorem of Arithmetic)

HCF (Highest Common Factor) — Definition and Methods

The HCF of two or more numbers is the largest number that divides each of them exactly. It is also called the Greatest Common Divisor (GCD). Three methods are taught in Class 6 Mathematics solutions: listing common factors, prime factorisation and the continued-division (or Euclidean) method. For small numbers, list all factors and pick the largest common one. For larger numbers, write the prime factorisation of each, identify common prime factors, then multiply the lowest power of each common prime. The continued-division method (dividing both numbers by common primes step-by-step) is faster for two numbers but requires practice. HCF is used to simplify fractions, solve word problems on grouping and find common periods in cyclical events. Remember: HCF is always less than or equal to the smallest of the given numbers.
  • HCF by listing — write all factors, circle common ones, pick the largest
  • HCF by prime factorisation — multiply the lowest powers of common primes
  • HCF by continued division — divide both by common primes until no common factor remains except 1
  • HCF(a,b) ≤ min(a,b); if HCF=1, numbers are co-prime

LCM (Lowest Common Multiple) — Definition and Methods

The LCM of two or more numbers is the smallest number that is a multiple of each. LCM is essential for adding or subtracting fractions with different denominators and solving time-based problems (bells ringing together, buses departing simultaneously). Three methods mirror those for HCF: listing multiples, prime factorisation and continued division. Listing is tedious for large numbers. Prime factorisation is most reliable: write each number as a product of primes, then multiply the highest power of every prime that appears. The continued-division method for LCM divides by any common or uncommon prime, carries down non-divisible numbers and stops when all quotients are 1; the product of all divisors is the LCM. A useful check is the formula HCF(a,b)×LCM(a,b)=a×b for two numbers. LCM is always greater than or equal to the largest of the given numbers.
  • LCM by listing — write multiples of each, find the smallest common one
  • LCM by prime factorisation — multiply the highest powers of all primes present
  • LCM by continued division — divide by primes (common or not) until all quotients are 1; multiply divisors
  • Formula: HCF×LCM = Product of the two numbers (for two numbers only)
  • LCM(a,b) ≥ max(a,b)

Sieve of Eratosthenes and Finding Primes

The Sieve of Eratosthenes is an ancient Greek algorithm to list all prime numbers up to a given limit, introduced in NCERT Class 6 Mathematics Chapter 5 Prime Time as a systematic method rather than trial division. Write all numbers from 2 to the limit. Circle 2 (the first prime) and cross out all its multiples (4,6,8…). Move to the next uncrossed number (3), circle it and cross out its multiples (6,9,12…). Repeat with 5, then 7 and so on. Stop when you reach the square root of the limit; all remaining uncrossed numbers are prime. This sieve is efficient for finding many primes at once and demonstrates that primes thin out as numbers grow larger. Between 1 and 10 there are four primes; between 91 and 100 only one (97). Memorise the first ten primes: 2,3,5,7,11,13,17,19,23,29. The sieve also helps verify whether a specific number is prime by checking divisibility only by primes up to its square root.
  • Write numbers 2 to N; circle 2, cross out all multiples of 2
  • Circle next uncrossed number (3), cross out its multiples
  • Repeat for 5,7,11… up to √N
  • All uncrossed numbers are prime
  • First ten primes: 2,3,5,7,11,13,17,19,23,29

Memory Tricks and Mnemonics

Chapter 5 Prime Time contains many rules and definitions that students mix up under exam pressure. Use these mnemonics and visual cues to lock them in memory. For divisibility by 3 and 9, remember 'Add digits: 3 if sum ÷3, 9 if sum ÷9'. For 11, picture a seesaw: odd-place digits on one side, even-place on the other; balance (difference zero or ÷11) means divisible. HCF stands for 'Highest Common Factor' — think 'common and small'; LCM is 'Lowest Common Multiple' — think 'common and large'. To recall that HCF uses lowest powers and LCM uses highest powers, associate H in HCF with 'humble' (lowest) and L in LCM with 'luxury' (highest). For prime factorisation, start with the smallest prime (2) and work upward; write divisors in ascending order to avoid missing factors. Co-prime does not mean both numbers are prime; it means their HCF is 1. Visualise co-prime as 'cooperating primes' that share nothing but 1. These memory aids appear repeatedly in Class 6 Mathematics notes and cut revision time before exams.
  • Divisibility by 3/9 — 'Add All Digits' mnemonic
  • Divisibility by 11 — 'Odd minus Even' seesaw balance
  • HCF mnemonic — 'Humble powers' (lowest)
  • LCM mnemonic — 'Luxury powers' (highest)
  • Co-prime — 'HCF = 1', not 'both are prime'
  • Prime factorisation — always start with 2, move up

Common Mistakes, Notation and Unit Pitfalls

Students lose marks in CBSE Class 6 Mathematics Chapter 5 tests by confusing HCF with LCM, forgetting that 1 is neither prime nor composite, and misapplying divisibility rules. A frequent error is writing 1 as a prime factor; 1 is excluded from prime factorisation. Another mistake is assuming co-prime numbers must themselves be prime; 8 and 9 are co-prime yet both composite. When using the test for 11, students often add instead of subtract or reverse the order (even−odd instead of odd−even); always subtract sum of even places from sum of odd places. In HCF by prime factorisation, taking the highest powers instead of lowest yields the LCM. Notation errors include omitting exponents (writing 2·2·3·3 instead of 2²×3²) and forgetting the multiplication sign between primes. Units are not applicable in Prime Time (numbers are abstract), but in word problems always state the final answer with context—'12 students per group' not just '12'. Double-check your factor tree ends with all prime leaves before multiplying. These pitfalls are easily avoided with careful reading and step-by-step working.
  • Never write 1 in prime factorisation; 1 is not prime
  • Co-prime ≠ both numbers are prime; it means HCF=1
  • Test for 11: (odd places) − (even places), not the reverse
  • HCF uses lowest powers; LCM uses highest powers—do not swap
  • Always write exponents in final prime factorisation (2³ not 2·2·2)
  • In word problems, include units or context in the final answer

Three Solved Mini-Examples Applying Formulas

Example 1: Find the HCF and LCM of 24 and 36 using prime factorisation. Solution: 24=2³×3, 36=2²×3². Common primes: 2,3. HCF=2²×3=12 (lowest powers). All primes: 2,3. LCM=2³×3²=72 (highest powers). Verify: 12×72=864=24×36 ✓. Example 2: Test if 2024 is divisible by 11. Solution: Digits of 2024 are 2,0,2,4. Odd places (from right): positions 1,3 → digits 4,2 → sum=6. Even places: positions 2,4 → digits 2,0 → sum=2. Difference: 6−2=4. Since 4 is not 0 or divisible by 11, 2024 is not divisible by 11. Example 3: Express 180 in prime factorisation form. Solution: 180÷2=90, 90÷2=45, 45÷3=15, 15÷3=5, 5÷5=1. Primes: 2,2,3,3,5. So 180=2²×3²×5. These worked examples cover the three pillars of Prime Time: HCF and LCM calculation, divisibility testing and prime factorisation. Practise similar problems from CBSE Class 6 Mathematics solutions to gain speed and accuracy.

One-Glance Last-Minute Revision Box

Use this summary table for quick revision the night before your exam. Prime: exactly two factors. Composite: more than two factors. 1 is neither. Divisibility: 2→even last digit; 3→digit-sum÷3; 5→last digit 0 or 5; 9→digit-sum÷9; 10→last digit 0; 11→(odd-place sum)−(even-place sum)÷11. Prime factorisation: factor tree or repeated division; write answer with exponents. HCF: product of lowest powers of common primes; HCF≤smaller number. LCM: product of highest powers of all primes; LCM≥larger number. Formula: HCF×LCM=product of two numbers. Co-prime: HCF=1. Sieve of Eratosthenes: circle and eliminate multiples to list primes. First ten primes: 2,3,5,7,11,13,17,19,23,29. Keep this box on your desk during practice tests. For deeper explanations, video solutions and instant doubt clearing by uploading a photo of any problem, students across India rely on CBSETUTOR.ai—one flat fee of ₹999 per month for all classes 6 to 12, with a 3-day free trial so your child can experience 24×7 AI tutoring before committing. Master Prime Time formulas with confidence and watch your accuracy soar in school assessments and board exams.
  • Prime = 2 factors; Composite = >2 factors; 1 = neither
  • Divisibility shortcuts: memorise tests for 2,3,5,9,10,11
  • Prime factorisation: always express with exponents (2³×3²)
  • HCF: lowest powers of common primes; LCM: highest powers of all primes
  • HCF×LCM = product (for two numbers)
  • Sieve: eliminate multiples systematically to find all primes up to N

Frequently asked questions

What is the difference between prime and composite numbers?+
A prime number has exactly two distinct factors: 1 and the number itself (e.g. 2,3,5,7,11). A composite number has more than two factors (e.g. 4,6,8,9). The number 1 is special—it is neither prime nor composite.
Why is 1 neither prime nor composite?+
By definition, a prime must have exactly two factors. The number 1 has only one factor (itself), so it does not meet the criterion. Similarly, composites have more than two factors, which 1 does not. Hence 1 sits in a category of its own as the unit.
How do I quickly check if a large number is divisible by 11?+
Add the digits at odd positions (1st,3rd,5th… from the right), then add the digits at even positions (2nd,4th,6th…). Subtract the even sum from the odd sum. If the result is 0 or divisible by 11, the original number is divisible by 11.
What is the Sieve of Eratosthenes and when should I use it?+
The Sieve of Eratosthenes is an algorithm to list all primes up to a limit. Write numbers from 2 upward, circle the first uncrossed number (a prime) and cross out all its multiples. Repeat until you reach the square root of the limit. Use it when you need many primes at once, not just one prime factorisation.
Should I use factor trees or the division method for prime factorisation?+
Both yield the same result; choose whichever feels clearer. Factor trees are visual and intuitive for beginners. The division method is faster for larger numbers once you are comfortable with the sequence of primes (2,3,5,7…). Practice both in Class 6 Mathematics solutions.
How do I find HCF using prime factorisation?+
Write the prime factorisation of each number. Identify the common prime factors. Multiply together the lowest power of each common prime. That product is the HCF. For example, HCF of 2²×3 and 2³×3² is 2²×3=12.
How do I find LCM using prime factorisation?+
Write the prime factorisation of each number. List every prime that appears in any factorisation. Multiply together the highest power of each prime. That product is the LCM. For 2²×3 and 2³×5, LCM is 2³×3×5=120.
What does it mean if two numbers are co-prime?+
Two numbers are co-prime (or relatively prime) if their HCF is 1—they share no common prime factor. For example, 8 and 15 are co-prime even though 8=2³ and 15=3×5 are both composite. Co-prime does not mean both numbers must be prime.
Can I use the formula HCF × LCM = Product for more than two numbers?+
No, the formula HCF(a,b)×LCM(a,b)=a×b holds only for exactly two numbers. For three or more numbers, compute HCF and LCM separately using prime factorisation or continued division; no single product formula exists.
Why does CBSE emphasise divisibility tests in Class 6 Mathematics Chapter 5?+
Divisibility tests build mental-maths fluency and speed up HCF, LCM and fraction simplification without a calculator. They reappear in algebra (factorisation), ratio-proportion and higher number theory, making them foundational skills for secondary mathematics and competitive exams.

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