KJSEA 2025 Maths Q1 — Prime Numbers (Grade 9)

KJSEA 2025 Grade 9 Numbers

Published

The Question

“Sally picked number cards showing 2, 4, 5 and 9. Which of the sets has prime numbers only?”

1

Recall what a prime number is

Start by fixing the rule you will test each card against. A prime number is a whole number that has exactly two different factors: the number one and the number itself, and nothing else divides into it evenly. Any number with more factors than that is not prime, so this single definition decides every card.

2

Test the card showing 2

Check which numbers divide exactly into two. Only one and two go into it, which is exactly two factors, so two fits the definition and is prime. It is worth noting that two is the only even prime number.

2=1×2⇒factors: 1,2    (prime)2 = 1 \times 2 \Rightarrow \text{factors: } 1, 2 \;\;(\text{prime})
3

Test the card showing 4

Look for the factors of four. Besides one and four it can also be made from two times two, so it has an extra factor of two. Because it has more than two factors, four is not prime.

4=2×2⇒factors: 1,2,4    (not prime)4 = 2 \times 2 \Rightarrow \text{factors: } 1, 2, 4 \;\;(\text{not prime})
4

Test the card showing 5

Check what divides into five. Only one and five divide it exactly, which is again exactly two factors, so five is prime.

5=1×5⇒factors: 1,5    (prime)5 = 1 \times 5 \Rightarrow \text{factors: } 1, 5 \;\;(\text{prime})
5

Test the card showing 9

Find the factors of nine. As well as one and nine, it is also three times three, giving an extra factor of three. With more than two factors, nine is not prime.

9=3×3⇒factors: 1,3,9    (not prime)9 = 3 \times 3 \Rightarrow \text{factors: } 1, 3, 9 \;\;(\text{not prime})
6

Collect the primes

Gather the cards that passed the two-factor test. Only two and five had exactly two factors each, so the set that contains prime numbers only is the set holding two and five.

{ 2,  5 }\{\,2,\;5\,\}

Final Result

The set with prime numbers only is 2 and 5, because these are the only cards whose factors are just one and the number itself.

Why this method works

The method works because being prime is defined entirely by counting factors, not by whether a number looks big or small. If a number can be written as a product of two whole numbers other than one and itself, that product reveals an extra factor and rules it out as prime. Four and nine are both perfect squares of a smaller whole number, so each gains that middle factor and fails the test, while two and five cannot be broken down any further and stay prime.

✓

Two and five each have exactly two factors; four and nine each have three factors, confirming only 2 and 5 are prime.