Primes & Prime Factorization • Topic 1 of 4

Prime & Composite Numbers

A prime number has exactly two positive factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 — worth memorising through at least 30. Two facts trip people up: 1 is not prime (it has only one factor), and 2 is the only even prime (every other even number has 2 as a third factor). A composite number has more than two factors. To test whether n is prime, you only need to try prime divisors up to √n — if none divides n, it is prime; for example, to check 97 you test 2, 3, 5, 7 (since 7² = 49 < 97 < 121 = 11²) and stop. On the GRE, "prime" is a favourite constraint in Quantitative Comparison because it quietly forces 2 into the picture as the lone even case that can break a pattern.

✅ Solved examples

1. Is 91 prime?
Test primes up to √91 ≈ 9.5: 2 (no, odd), 3 (digit sum 10, no), 5 (no), 7 → 7 × 13 = 91. It divides, so 91 is composite, not prime.
2. How many prime numbers are there between 20 and 40?
Check odds not divisible by 3 or 5: 23, 29, 31, 37 are prime (21, 25, 27, 33, 35, 39 are composite). That is 4 primes.
3. p is a prime greater than 2. Is p odd or even?
2 is the only even prime; every prime larger than 2 must be odd (any larger even number is divisible by 2). So p is odd.
4. Quantitative Comparison. p is a prime number. Quantity A: the number of positive factors of p. Quantity B: 2. Which is greater?
By definition a prime has exactly two factors, 1 and itself. Quantity A = 2 = Quantity B; they are equal.

✏️ Practice — try these, take hints as needed

1. Is 51 prime?
Digit sum 6.
Divisible by 3.
3 × 17.
No (composite)
2. List the primes between 40 and 50.
Odd, not by 3 or 5.
Check 41, 43, 47.
49 = 7².
41, 43, 47
3. What is the only even prime number?
Every other even has factor 2.
Smallest prime.
It is 2.
2
4. Is 1 a prime number?
Primes have two factors.
1 has one factor.
So no.
No
5. Is 143 prime?
Try small primes.
11 × 13.
Check the product.
No (11 × 13)

📝 Topic test — 8 questions

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