Factors and Multiples • Topic 1 of 7

Prime Numbers

A prime number is a whole number greater than 1 with exactly two distinct factors: 1 and itself. The primes begin 2, 3, 5, 7, 11, 13, 17, 19, 23, … . Two is the only even prime — every other even number is divisible by 2, giving it a third factor. The number 1 is not prime (it has only one factor) and not composite. To test whether n is prime, check for divisibility by each prime up to √n; if none divides n, it is prime. On the SAT, primes appear in factoring, in “how many primes” counting, and in clever divisibility arguments.

✅ Solved examples

1. Is 91 a prime number?
Test primes up to √91 ≈ 9.5, so check 2, 3, 5, 7. It is odd (not ÷2), digit sum 9+1=10 (not ÷3), does not end in 0/5 (not ÷5). But 91 = 7 × 13, so 7 divides it. Therefore 91 is NOT prime — it is composite.
2. How many prime numbers are there between 20 and 40?
List candidates and remove composites: 23 (prime), 29 (prime), 31 (prime), 37 (prime). The numbers 21=3×7, 25=5², 27=3³, 33=3×11, 35=5×7, 39=3×13 are composite. So there are 4 primes: 23, 29, 31, 37.
3. Why is 2 the only even prime number?
A prime has exactly two factors. Every even number is divisible by 2. For an even number greater than 2, the divisors include 1, 2 and the number itself — at least three factors — so it cannot be prime. Only 2 has exactly the two factors 1 and 2, making it the single even prime.
4. The sum of the first five prime numbers is what?
The first five primes are 2, 3, 5, 7, 11 (remember to start at 2, and that 1 is not prime). Their sum is 2 + 3 + 5 + 7 + 11 = 28.

✏️ Practice — try these, take hints as needed

1. Is 143 a prime number?
Find √143 ≈ 11.9, so test the primes 2, 3, 5, 7, 11.
It is odd, digit sum 8 (not ÷3), and does not end in 0 or 5.
Try 11: 143 = 11 × 13.
No — 143 = 11 × 13 is composite.
2. How many primes lie between 1 and 20?
Remember 1 is not prime; start counting from 2.
List 2, 3, 5, 7, then the teens: 11, 13, 17, 19.
Count them all together.
8 primes (2, 3, 5, 7, 11, 13, 17, 19).
3. p is a prime number and p + 2 is also prime. Give all such p below 10.
These are called twin primes — pairs differing by 2.
Test each prime: 3→5, 5→7, 7→9.
Check whether p+2 is itself prime in each case (9 = 3×3 is not).
p = 3 (3, 5) and p = 5 (5, 7).
4. What is the largest prime number less than 50?
Work downward from 49.
49 = 7², 48 and 46 are even, 45 ÷ 5.
Test 47 for divisibility by 2, 3, 5 — none divide it.
47.
5. The product of two primes is 35. What are the two primes?
35 is odd, so neither prime is 2.
Factor 35 into a product of two numbers.
35 = 5 × 7, and both 5 and 7 are prime.
5 and 7.

📝 Topic test — 8 questions

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