Exercise 1.2 — Fundamental Theorem

What Does Exercise 1.2 Cover?

Exercise 1.2 of Class 10 Mathematics — Chapter 1: Real Numbers — focuses on the Fundamental Theorem of Arithmetic and its applications. The theorem states that every composite number can be expressed as a product of prime numbers in exactly one way (the order of factors may differ, but the set of prime factors is unique). This exercise teaches you to use prime factorisation to find HCF and LCM efficiently, and to reason about numbers using their prime structure — skills tested in CBSE, Telangana, and Andhra Pradesh board exams.

The Fundamental Theorem of Arithmetic

No matter who factorises a composite number or in what order they divide, the final prime factors are always the same. For example, 180 can be broken down starting from 2, 3, or 5 — but every path leads to the same result:

This uniqueness is what makes prime factorisation so powerful. It gives every composite number a kind of "fingerprint" that you can use to compare numbers, find common factors, and calculate LCM with precision.

Question 1 — Prime Factorisation of Numbers

The first question asks you to express five numbers as products of their prime factors using factor trees or repeated division. Here are the results in exponential form:

• 140 = 2² × 5 × 7
• 156 = 2² × 3 × 13
• 3825 = 3² × 5² × 17
• 5005 = 5 × 7 × 11 × 13 (all prime factors appear once)
• 7429 = 17 × 19 × 23 (all prime factors appear once)

For numbers like 5005 and 7429, students often try to divide by small primes like 2 or 3 first and waste time. Since these numbers are not even and their digit sums are not divisible by 3, start testing from 5 or 7 directly.

Question 2 — Finding LCM and HCF by Prime Factorisation

Once you have the prime factorisation of each number, the rules are straightforward: the HCF uses the smallest power of each common prime factor, and the LCM uses the greatest power of every prime factor present across all numbers.

• 12, 15, 21: Only prime factor common to all three is 3¹. HCF = 3; LCM = 2² × 3 × 5 × 7 = 420.
• 17, 23, 29: All three are primes with no common factor. HCF = 1; LCM = 17 × 23 × 29 = 11339.
• 8, 9, 25: No prime factor is shared among all three (8 = 2³, 9 = 3², 25 = 5²). HCF = 1; LCM = 1800.
• 72 and 108: 72 = 2³ × 3², 108 = 2² × 3³. HCF = 2² × 3² = 36; LCM = 2³ × 3³ = 216.
• 306 and 657: 306 = 2 × 3² × 17, 657 = 3² × 73. HCF = 3² = 9; LCM = 22338.

A useful verification: for any two numbers, HCF × LCM = Product of the two numbers. Check your answers with this relationship every time.

Question 3 — Can 6ⁿ Ever End in Zero?

This is a classic reasoning question. A number ends in zero only if its prime factorisation contains both 2 and 5 as factors (since 10 = 2 × 5). Now, 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ. There is no factor of 5 anywhere in this expression, regardless of what n is. Since 5 is never a prime factor of 6ⁿ, it can never produce a 10 as a factor, and therefore 6ⁿ cannot end with the digit 0 for any natural number n.

Questions 4 & 5 — Proving Expressions Are Composite

These questions test whether you can factor out a common term to reveal that a number has more than two factors — the definition of a composite number.

• 7 × 11 × 13 + 13: Factor out 13 to get 13 × (77 + 1) = 13 × 78. Since it has factors 13 and 78 beyond 1 and itself, it is composite.
• 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5: Factor out 5 to get 5 × (1008 + 1) = 5 × 1009. Both 5 and 1009 are factors, so it is composite.
• 17 × 11 × 2 + 17 × 11 × 5: Factor out 17 × 11 to get 17 × 11 × (2 + 5) = 17 × 11 × 7. Having three prime factors immediately confirms it is composite.

Question 6 — Last Digit of 6¹⁰⁰

Every power of 6 ends in 6: 6¹ = 6, 6² = 36, 6³ = 216, and so on. This pattern holds for any natural number exponent because 6 × 6 always gives a number ending in 6. Therefore, the last digit of 6¹⁰⁰ is 6. This is a direct consequence of the prime structure of 6 and is a frequently asked one-mark question in board exams.

Common Mistakes in Exercise 1.2

• Using the largest power for HCF instead of the smallest — remember: HCF takes the minimum power of common factors, LCM takes the maximum.
• Forgetting to include prime factors that appear in only one number when computing LCM — LCM must account for every prime that appears in any of the numbers.
• In Questions 4 and 5, trying to evaluate the full expression numerically instead of factoring — always look for a common factor to pull out first.
• Skipping the HCF × LCM = Product verification step, which is the fastest way to catch errors before submitting.

What to Study Next

With the Fundamental Theorem of Arithmetic mastered, the next step is understanding how to prove that numbers like √2, √3, and √5 are irrational — proofs that rely directly on prime factorisation logic. You can also revisit how HCF was found a different way in Exercise 1.1 using Euclid's Division Algorithm, and see how both methods connect back to the Real Numbers Introduction.