Fundamental Theorem of Arithmetic
- CBSE · NCERT Class 10, Ch 1 Real Numbers
The Fundamental Theorem of Arithmetic says every composite number can be written as a product of primes in exactly one way, apart from the order.
A prime number has exactly two factors, 1 and itself. A composite number has more. The theorem says you can always break a composite number into primes. It also says the result is always the same. You may split it in a different order, but you end with the same primes, each used the same number of times.
A small example
Take 360. One route is 360 = 36 × 10. Another is 360 = 8 × 45. Keep splitting, and both routes end in the same place: 360 = 2 × 2 × 2 × 3 × 3 × 5, or 2³ × 3² × 5. No other set of primes multiplies to 360.
This is why the theorem is useful. A number's primes tell you facts about it. A number ends in 0 only if both 2 and 5 are among its prime factors. So 25ⁿ, which is 5²ⁿ, can never end in 0. The prime 2 is missing.
Where marks go
Question 21(B) of CBSE's 2026-27 Maths Basic sample paper asks exactly this, for 2 marks. The board's Maths Basic marking scheme for 2026-27 splits it into four half marks:
- Writing 25ⁿ as 5²ⁿ, a power of 5 only.
- Saying that a number ending in 0 needs both 2 and 5 as prime factors.
- Noting that 2 is not a factor of 25ⁿ.
- The conclusion, that 25ⁿ never ends in 0.
So the conclusion alone is worth half a mark. The other three half marks are for the reasons. This is step marking at work: the examiner looks for each step the scheme lists.
Sources
Facts last checked against these sources on 30 September 2026.
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