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CBSE · Class 10 · Mathematics

Real Numbers 1.2: The Fundamental Theorem of Arithmetic

5 concepts · 40 practice questions with answers · Free to read

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Prime factorisation and uniqueness

From daily life

Every LEGO model is built from the same small bricks. A number is built from prime "bricks" in just one way.

Simple definition

The Fundamental Theorem of Arithmetic: every composite number can be written as a product of primes, and this factorisation is unique apart from the order of the factors.

Formula: n = p₁^a × p₂^b × p₃^c …

602302153560 = 2² × 3 × 5

In the figure: The factor tree of 60 ends in the primes 2, 2, 3, 5, whatever order you split it in.

Live figure

Prime factorisation and uniqueness

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Rule 1: Build the factor tree — Keep splitting until every branch is a prime, then write repeated primes as powers.

  • Easy: 60 = 2² × 3 × 5.
  • Medium: 588 = 2² × 3 × 7².
  • Exam-type: 1 is not a prime and not a composite.

Rule 2: Uniqueness — Two different trees for the same number give the same primes.

  • Easy: 36 = 4 × 9 = 6 × 6 → 2² × 3² both ways.
  • Medium: If 2 × 3 × 5 = n then n = 30 and no other primes can appear.
  • Exam-type: Check by multiplying back: 2² × 3² × 5 = 180.
Remember

Always finish with powers of primes in increasing order, for example 2² × 3 × 5.

Common mistake

Stopping while a composite factor like 4 or 9 is still left in the product.

Memory clue

"Split till every branch is a prime."

Practice questions

Easy
  1. 1How many different prime factors does 40 have?
  2. 2Which is the smallest prime number?
  3. 3Express 3375 as a product of prime factors.
Medium
  1. 4Express 1764 as a product of prime factors.
  2. 5How many different prime factors does 15435 have?
  3. 6How many different prime factors does 5324 have?
Challenging
  1. 7In the prime factorisation of 1000, what is the exponent of the prime 5?
  2. 8In the prime factorisation of 784, what is the exponent of the prime 7?
Show answers (try them yourself first)
  1. 12
  2. 22
  3. 33³ × 5³
  4. 42² × 3² × 7²
  5. 53
  6. 62
  7. 73
  8. 82

HCF by prime factorisation

From daily life

Two ropes of 12 m and 18 m are cut into equal pieces with no waste. The longest piece is their HCF.

Simple definition

HCF = product of the common primes, each with the smallest power it has in the numbers.

Formula: HCF = ∏ (common primes)^(least power)

12 = 2² × 318 = 2 × 3²smallest powers2¹ × 3¹ = 6HCF(12, 18) = 6common primes, lowest power

In the figure: HCF(12, 18): common primes 2 and 3 at their lowest powers give 2 × 3 = 6.

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HCF by prime factorisation

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Rule 1: Two numbers — Factorise both, then keep only primes present in both.

  • Easy: HCF(36, 60): 2² × 3 = 12.
  • Medium: HCF(96, 404) = 4.
  • Exam-type: HCF(8, 9) = 1 (no common prime): co-prime.

Rule 2: Three numbers — Take the primes common to all three.

  • Easy: HCF(12, 18, 24) = 2 × 3 = 6.
  • Medium: HCF(8, 12, 20) = 2² = 4.
  • Exam-type: Use HCF to make the largest equal groups.
Remember

HCF divides every number; it can never be bigger than the smallest of them.

Common mistake

Taking the highest power, or primes that are not common to all numbers.

Memory clue

"HCF: common and lowest."

Practice questions

Easy
  1. 1If a = 2² × 7 and b = 2³ × 3 × 7³, then HCF(a, b) = ?
  2. 2If a = 2 × 5 and b = 2³ × 3³ × 5³, then HCF(a, b) = ?
  3. 3Find the HCF of 20 and 24 using prime factorisation.
Medium
  1. 4Find the HCF of 12 and 84 using prime factorisation.
  2. 5In the prime factorisation method, the HCF is found by taking the ___ power of each COMMON prime.
  3. 6Find the largest number that divides 151 and 164 leaving remainders 15 and 11 respectively.
Challenging
  1. 7Find the largest number that divides 52 and 21 leaving remainders 2 and 1 respectively.
  2. 8Find the HCF of 8, 4 and 10.
Show answers (try them yourself first)
  1. 12² × 7
  2. 22 × 5
  3. 34
  4. 412
  5. 5smallest
  6. 617
  7. 710
  8. 82

LCM by prime factorisation

From daily life

Two lights blink every 12 and 18 seconds. They blink together again after their LCM.

Simple definition

LCM = product of all the primes that appear, each with its greatest power.

Formula: LCM = ∏ (all primes)^(greatest power)

12 = 2² × 318 = 2 × 3²all primes, largest powers2² × 3² = 36LCM(12, 18) = 36every prime, highest power

In the figure: LCM(12, 18): 2² and 3² give 4 × 9 = 36.

Rule 1: LCM of two numbers — Collect every prime, with its highest power.

  • Easy: LCM(6, 20) = 2² × 3 × 5 = 60.
  • Medium: LCM(8, 9) = 72.
  • Exam-type: LCM(12, 15, 21) = 2² × 3 × 5 × 7 = 420.

Rule 2: Meeting-again problems — "After how long together again?" asks for the LCM.

  • Easy: Beats every 12 s and 18 s: together after 36 s.
  • Medium: Two runners lap in 24 and 36 min: meet after 72 min.
  • Exam-type: LCM is never smaller than the largest number.
Remember

HCF takes the common primes (lowest power); LCM takes all primes (highest power).

Common mistake

Mixing the two rules: using the lowest powers for LCM.

Memory clue

"LCM: all and highest."

Practice questions

Easy
  1. 1If a = 3 × 7 × 11 and b = 7² × 11², then LCM(a, b) = ?
  2. 2If a = 3³ × 5 and b = 3³ × 5², then LCM(a, b) = ?
  3. 3Find the LCM of 120 and 84 using prime factorisation.
Medium
  1. 4What is the smallest positive number that is exactly divisible by both 12 and 40?
  2. 5Find the LCM of 20 and 50 using prime factorisation.
  3. 6In the prime factorisation method, the LCM is found by taking the ___ power of EVERY prime that appears.
Challenging
  1. 7Find the LCM of 16, 25 and 24.
  2. 8Find the smallest number which, when divided by 20, 8 and 18, leaves remainder 7 each time.
Show answers (try them yourself first)
  1. 13 × 7² × 11²
  2. 23³ × 5²
  3. 3840
  4. 4120
  5. 5100
  6. 6greatest
  7. 71200
  8. 8367

HCF × LCM = product of the two numbers

From daily life

If you know the HCF and LCM, you can find a missing number without factorising it.

Simple definition

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. This works for two numbers only, not three.

Formula: HCF × LCM = a × b

6 × 36 = 216=12 × 18 = 216HCF × LCM = a × b (two numbers only)

In the figure: 12 and 18: HCF 6, LCM 36, and 6 × 36 = 216 = 12 × 18.

Rule 1: Find the missing number — b = HCF × LCM ÷ a.

  • Easy: HCF 9, LCM 360, one number 45: other = 9 × 360 ÷ 45 = 72.
  • Medium: HCF 4, LCM 84, one number 12: other = 28.
  • Exam-type: Co-prime numbers: LCM = product.

Rule 2: Find the LCM or HCF — Use whichever one is missing.

  • Easy: HCF(25, 35) = 5 so LCM = 875 ÷ 5 = 175.
  • Medium: HCF(8, 9) = 1 so LCM = 72.
  • Exam-type: For three numbers this rule fails.
Remember

The HCF always divides the LCM.

Common mistake

Using HCF × LCM = a × b × c for three numbers.

Memory clue

"Two numbers: product = HCF × LCM."

Practice questions

Easy
  1. 1For two positive integers a and b, HCF(a, b) × LCM(a, b) = ?
  2. 29 and 11 are co-prime numbers. What is their LCM?
  3. 326 and 29 are co-prime numbers. What is their LCM?
Medium
  1. 425 and 23 are co-prime numbers. What is their LCM?
  2. 5The HCF of two numbers is 11, their LCM is 968, and one of the numbers is 88. Find the other number.
  3. 6The HCF of two numbers is 13, their LCM is 65, and one of the numbers is 13. Find the other number.
Challenging
  1. 7The product of two positive integers is 256 and their HCF is 8. What is their LCM?
  2. 8Two numbers are in the ratio 11 : 2 and their HCF is 3. Find their LCM.
Show answers (try them yourself first)
  1. 1a × b
  2. 299
  3. 3754
  4. 4575
  5. 5121
  6. 665
  7. 732
  8. 866

Using the theorem

From daily life

Why does 10 end in 0? Because it has both the primes 2 and 5. The theorem explains many such facts.

Simple definition

A number ends in 0 only if its prime factorisation has both 2 and 5. A number is composite if it has a factor other than 1 and itself. If n = p^a × q^b, it has (a + 1)(b + 1) factors.

Formula: factors = (a + 1)(b + 1)

ends in 0 ⇒ 2 and 56ⁿ has no 5 ⇒ never 036 = 2² × 3²(2+1)(2+1) = 9 factorsuse the prime factors

In the figure: 6ⁿ = 2ⁿ × 3ⁿ has no prime 5, so it never ends in 0.

Rule 1: Ending digit and composite checks — Look for the primes 2 and 5, or a common factor.

  • Easy: 4ⁿ = 2²ⁿ has no 5, never ends in 0.
  • Medium: (2 × 5 × 7) × 3 + 5 has a factor 5 so it is composite.
  • Exam-type: 7 × 11 × 13 + 13 = 13 × (77 + 1) = 13 × 78: composite.

Rule 2: Number of factors — Add 1 to each power and multiply.

  • Easy: 36 = 2² × 3²: 3 × 3 = 9 factors.
  • Medium: 60 = 2² × 3 × 5: 3 × 2 × 2 = 12 factors.
  • Exam-type: A prime has exactly 2 factors.
Remember

A number is a perfect square if every prime power in its factorisation is even.

Common mistake

Counting factors as a + b instead of (a + 1)(b + 1).

Memory clue

"Add one to each power, then multiply."

Practice questions

Easy
  1. 1A composite number has at least ___ prime factors when repeated factors are counted separately.
  2. 2How many positive factors does 891 have (including 1 and 891)?
  3. 3How many positive factors does 5488 have (including 1 and 5488)?
Medium
  1. 4How many positive factors does 12 have (including 1 and 12)?
  2. 5784 = 2⁴ × 7². Is 784 a perfect square?
  3. 6How many positive factors does 676 have (including 1 and 676)?
Challenging
  1. 71225 = 5² × 7². Is 1225 a perfect square?
  2. 8How many positive factors does 1125 have (including 1 and 1125)?
Show answers (try them yourself first)
  1. 12
  2. 210
  3. 320
  4. 46
  5. 5Yes, every exponent is even
  6. 69
  7. 7Yes, every exponent is even
  8. 812