Prime Numbers

The definition, and two common exceptions

A prime number is a natural number greater than 1 whose only positive divisors are 1 and itself. A number greater than 1 that isn't prime is composite.

0 is neither prime nor composite: the two words are only defined for natural numbers greater than 1. (0 couldn't pass the test anyway: every non-zero number divides 0, so it has far more than two divisors.)

1 is also neither: a prime needs exactly two distinct divisors (1 and itself), but 1's only divisor is 1 itself, just one.

2 is the only even prime. Every other even number has 2 as a divisor as well as 1 and itself.

Prime numbers up to 100

There are 25 primes below 100:

Worked example: checking primality

Is $91$ prime?

Solution: It's enough to test divisibility by every prime up to $\sqrt{91} \approx 9.5$ (2, 3, 5, and 7), since any factor larger than the square root would have to pair with one smaller than it, already caught. $91$ is odd (not divisible by 2), its digit sum 10 isn't divisible by 3, it doesn't end in 0 or 5 (not divisible by 5), but $91 = 7 \times 13$, so 91 is composite, not prime.

Compare 97: $\sqrt{97} \approx 9.8$, and none of 2, 3, 5 or 7 divides it, so 97 is prime.

Prime factorization and distinct prime factors

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 of the factors. The primes that appear are its prime divisors (or prime factors); counted once each, they are its distinct prime factors.

Example: factorize 360 and list its distinct prime factors.

Solution: Keep dividing by the smallest prime that works: $360 = 2 \times 180 = 2^2 \times 90 = 2^3 \times 45 = 2^3 \times 3^2 \times 5$.

Its distinct prime factors are 2, 3 and 5: three of them, even though six primes multiply together to make 360. The exponents also give its number of divisors: $(3+1)(2+1)(1+1) = 24$.

Counting the factors of a number

Prime factorization also tells you how many factors a number has. Write $N = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$. Every factor of $N$ is made by choosing, for each prime, an exponent from 0 up to $a_i$, independently. So the number of factors, counting 1 and $N$ itself, is:

A small check. $36 = 2^2 \times 3^2$, so it has $(2+1)(2+1) = 9$ factors: 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Worked example. How many factors does 9600 have? Factorize first: $9600 = 2^7 \times 3^1 \times 5^2$. Add 1 to each exponent and multiply: $(7+1)(1+1)(2+1) = 8 \times 2 \times 3 = 48$.

Already factorized. $2^5 \cdot 3^6 \cdot 7^3$ has $(5+1)(6+1)(3+1) = 6 \times 7 \times 4 = 168$ factors. A prime that appears to the first power still contributes a factor of 2, as the 3 in 9600 does.

Twin primes

A pair of primes exactly 2 apart is called a twin prime pair, like $(17, 19)$ or $(29, 31)$. The very first few pairs are $(3,5), (5,7), (11,13), (17,19), (29,31), (41,43)$.

Co-prime numbers

Two numbers are co-prime when their HCF is 1, meaning they share no prime factor. Neither has to be prime itself: $8 = 2^3$ and $15 = 3 \times 5$ are co-prime. Any two different primes are always co-prime.

Common mistakes

• Calling 1 a prime. It has only one divisor. The smallest prime is 2.
• Trusting numbers that look prime. 51 ($3 \times 17$), 57 ($3 \times 19$) and 91 ($7 \times 13$) are the usual traps. Test every prime up to the square root.
• Stopping before the square root. You need every prime up to $\sqrt{n}$ and no further: 2, 3, 5 and 7 settle 97, but skipping 7 for 91 would miss its factor.
• Mixing up distinct and repeated prime factors. 360 has three distinct prime factors (2, 3, 5), but six prime factors counted with repetition.
• Thinking co-prime means both numbers are prime. 8 and 15 are co-prime and neither is prime.
• Multiplying the exponents instead of the exponents plus one when counting factors. $2^7 \times 3 \times 5^2$ has $8 \times 2 \times 3 = 48$ factors, not $7 \times 1 \times 2 = 14$.

Where this comes up in exams

The 2025-26 CBSE Class X syllabus opens its Real Numbers unit with the Fundamental Theorem of Arithmetic, the prime factorization result above.

Checked against CBSE Mathematics curriculum, Classes IX-X (2025-26). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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