Problem Set 35 – Co-prime Numbers and the SieveClass 5 Maths · Chapter 8 Multiples and Factors · Problem Set 35 – Maharashtra Board

Summary: Problem Set 35 – Co-prime Numbers and the Sieve

A common factor of two numbers is a number that divides both of them exactly. 1 is a common factor of every pair.

Two numbers are co-prime when 1 is their ONLY common factor. 9 and 16 are co-prime; 12 and 20 are not (2 and 4 also divide both).

Quick tests: two numbers next to each other (like 27 and 28) are always co-prime; two even numbers are never co-prime because 2 divides both.

Eratosthenes' sieve finds primes up to 100: keep 2, 3, 5, 7 and cross out their other multiples. The numbers left (except 1) are prime.

Twin primes are primes that differ by 2, like 11 and 13. There are 8 such pairs below 100, and Euclid showed that primes never run out.

Key points to remember

• Common factor = divides both numbers exactly
• Co-prime = only common factor is 1
• Co-prime numbers need not be prime themselves (8 and 15)
• Two even numbers are never co-prime
• Neighbours like 14 and 15 are always co-prime
• 1 is neither prime nor composite
• Sieve: cross out multiples of 2, 3, 5, 7 — the rest are primes
• Twin primes differ by 2: 3 & 5, 5 & 7, 11 & 13 …

Problem Set 35 – Co-prime Numbers and the Sieve – Notes

1. Common factors

• List the factors of each number, then look for the numbers that appear in both lists.
• Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 20: 1, 2, 4, 5, 10, 20.
• Common factors of 12 and 20: 1, 2 and 4 — three of them.

2. What co-prime means

• If the only number in both lists is 1, the two numbers are co-prime.
• 12 and 20 share 1, 2 and 4, so they are NOT co-prime.
• The numbers themselves can be composite: 8 (1, 2, 4, 8) and 15 (1, 3, 5, 15) share only 1, so they are co-prime.

3. Testing a pair step by step

• Step 1: write the factors of the first number.
• Step 2: write the factors of the second number.
• Step 3: circle the common ones. Only 1 → co-prime. More than 1 → not co-prime.
• Shortcut: if both are even, 2 is common, so stop — not co-prime.

4. Eratosthenes' sieve (primes up to 100)

• Eratosthenes was a Greek mathematician who lived around 250 BC.
• Write 1 to 100 in a grid. Put a square around 1 — it is neither prime nor composite.
• Circle 2 and cross out every other multiple of 2. Then circle 3 and cross out its multiples. Do the same with 5 and 7.
• Every number left uncrossed gets a circle: these are the primes. Crossed-out numbers are composite.
• Another trick: write numbers in 6 columns. Apart from 2 and 3, all primes fall in the columns that start with 1 and 5.

5. Twin primes and endless primes

• Twin primes are two primes with a gap of 2.
• The 8 twin pairs below 100: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).
• Euclid, another Greek mathematician (about 300 BC), proved the list of primes goes on forever.

Formulas at a glance

Key terms

Problem Set 35 – Co-prime Numbers and the Sieve – Flashcards

Read the question, answer it in your head, then tap the card to check. 10 cards.

Problem Set 35 – Co-prime Numbers and the Sieve – MCQ Quiz

10 questions. Tap an option — you will see at once if it is right and why.

How to study Problem Set 35 – Co-prime Numbers and the Sieve

⏱ About 25 minutes

1. Learn (5 min)
   Learn what a common factor is and what co-prime means.
2. Test (10 min)
   List factors for each pair and circle the common ones. Then make a 1–100 sieve.
3. Practise (10 min)
   Do the 4 'Try it yourself' sums, take the quiz and play 🎮 Play.