Why is 1 deliberately NOT counted as a prime number?

Class 10Real NumbersNumber Systems

ABecause allowing 1 as a prime would destroy the uniqueness of prime factorisation — 1 could be inserted any number of times
BBecause 1 is an even number
CBecause 1 is a composite number
DBecause 1 has more than two factors

Step-by-Step Solution

If 1 were prime, then

12 = 2^{2} \times 3 = 1 \times 2^{2} \times 3 = 1 \times 1 \times 2^{2} \times 3

and so on would all count as different prime factorisations, and the Fundamental Theorem of Arithmetic would fail. 1 is odd, is neither prime nor composite, and has exactly ONE factor — so the other three options are all factually wrong.

Answer: A) Because allowing 1 as a prime would destroy the uniqueness of prime factorisation — 1 could be inserted any number of times

Key idea: Fundamental Theorem of Arithmetic

**Theorem**: Every composite number can be expressed as a product of primes — UNIQUELY, up to the order of factors.

Example:

360 = 2^{3} \times 3^{2} \times 5

. There is no other set of primes whose product is 360.

**Consequences**:
1. **Finding HCF** by primes: HCF = product of LOWEST powers of common primes.
2. **Finding LCM** by primes: LCM = product of HIGHEST powers of all primes appearing.
3. **Identity**: HCF(a,b) × LCM(a,b) =

a \times b

.

Example:

a = 12 = 2^{2} \cdot 3, b = 18 = 2 \cdot 3^{2}

. HCF = 2·3 = 6, LCM =

2^{2} \cdot 3^{2} = 36

. Check:

6 \times 36 = 216 = 12 \times 18

. ✓

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