If p = {prime factors of 210} and Q = {prime less than 10}, find p $\cap$ Q

Step 1: Analyze the question We need the set P of prime factors of 210 and the set Q of primes less than 10. Then we find their intersection, which means the elements common to both sets.

Step 2: Concept

• A prime factor is a prime number that divides the number exactly.
• The intersection P ∩ Q contains only the elements that appear in both P and Q.

Step 3: Solution

Prime factorisation of 210: 210 = 2 × 105 = 2 × 3 × 35 = 2 × 3 × 5 × 7

So P = {2, 3, 5, 7}.

Primes less than 10: 2, 3, 5, 7, so Q = {2, 3, 5, 7}.

Intersection: P ∩ Q = {2, 3, 5, 7}

(Note that 1 is not a prime number, which rules out options A and C. Option B leaves out 7, which is both a factor of 210 and a prime below 10.)

Step 4: Answer The correct option is D. {2, 3, 5, 7}.

Tip: Remember that 1 is not prime, and the first four primes (2, 3, 5, 7) multiply to give 210. When P and Q turn out to be identical, their intersection is the same set.