Prime Time

1. Chapter at a Glance

Every counting number has its own "family" of factors (numbers that divide it exactly) and an endless line of multiples (numbers it divides exactly). Some numbers, called primes, have the smallest possible family — just 1 and themselves. Primes are the building blocks of all numbers, just as bricks build a house. In this chapter you will play number games with multiples, find common factors and common multiples, separate primes from composites, test divisibility quickly and break numbers into their prime factors.

2. Factors and Multiples

If a number a divides a number b exactly (with remainder 0), then a is a factor of b and b is a multiple of a. For example, 4 × 6 = 24, so 4 and 6 are factors of 24, and 24 is a multiple of both 4 and 6.

• 1 is a factor of every number. Every number is a factor of itself.
• Every factor of a number is less than or equal to the number. So a number has only a limited number of factors.
• Every number is a multiple of itself, and its multiples go on endlessly: 7, 14, 21, 28, …

Finding all factors in pairs

1. Start with 1 × the number.
2. Try 2, 3, 4, … in order; whenever one divides exactly, write the pair.
3. Stop when the pairs start repeating (when the first number passes the second).

3. Common Multiples and Common Factors

The Idli–Vada game

Children sit in a circle and count 1, 2, 3, … in turn. Whenever a number is a multiple of 3 they say "idli"; a multiple of 5, "vada"; a multiple of both 3 and 5, "idli-vada". The first "idli-vada" comes at 15, then 30, 45, … These are the common multiples of 3 and 5, and the smallest one, 15, is their lowest common multiple (LCM).

Jumping animals

A grasshopper jumps 3 steps at a time; a frog jumps 4 steps at a time, both starting at 0. The places both land on are common multiples of 3 and 4: 12, 24, 36, … The first meeting point is 12.

Common factors

Factors shared by two numbers are their common factors. The largest of them is the highest common factor (HCF).

4. Prime and Composite Numbers

• A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.
• A composite number has more than two factors. Examples: 4, 6, 8, 9, 10, 12.
• The number 1 has only one factor (itself), so it is neither prime nor composite.
• 2 is the only even prime — every other even number has 2 as an extra factor.

There are 25 primes between 1 and 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. The ancient Greek mathematician Euclid proved that there are infinitely many primes.

The Sieve of Eratosthenes

Eratosthenes, a Greek scholar (around 240 BCE), gave a clever way to find primes:

1. Write the numbers from 1 to 100 in a grid. Cross out 1.
2. Circle 2 (prime) and cross out all other multiples of 2.
3. Circle 3 and cross out all its other multiples.
4. Circle the next uncrossed number (5) and cross out its multiples; then do the same for 7.
5. All numbers left uncrossed are prime. (We can stop after 7 because 11 × 11 = 121 is already beyond 100.)

5. Co-prime Numbers

Two numbers are co-prime if their only common factor is 1 (their HCF is 1). They need not be primes themselves.

• 8 and 15: factors of 8 are 1, 2, 4, 8; of 15 are 1, 3, 5, 15 → only 1 is common → co-prime.
• Any two consecutive numbers (like 20 and 21) are always co-prime.
• Any two different primes are co-prime.
• Two even numbers can never be co-prime (both have factor 2).

6. Prime Factorisation

Writing a number as a product of primes is called its prime factorisation. Every composite number can be broken down into primes, and the set of primes is always the same, whichever way we start (only the order may differ).

Factor tree method

Split the number into any two factors, then keep splitting each factor until only primes remain.

Repeated division method

1. Divide by the smallest prime that divides the number (2, then 3, 5, 7, …).
2. Keep dividing the quotient until it becomes 1.
3. The divisors used, multiplied together, give the prime factorisation.

Using prime factorisation to test co-primes and divisibility

• Two numbers are co-prime if they have no common prime factor.
• A number divides another if all its prime factors (with repetition) appear in the other's factorisation. Is 45 a factor of 420? 45 = 3 × 3 × 5, but 420 has only one 3, so no.

7. Divisibility Tests

Why does the test for 4 work? Any number = (hundreds part) + (last two digits). Since 100 is divisible by 4, every hundreds part is divisible by 4, so only the last two digits matter. The same idea works for 8 with the last three digits because 1000 is divisible by 8.

8. Master Formula Table

9. Master Summary Table

Chart: number of primes in each block of 20

Primes become less frequent as numbers grow, but they never stop.

10. Mind Map

11. Tips and Tricks

12. Important One-liners

• 1 is a factor of every number.
• Every number has finitely many factors but infinitely many multiples.
• A prime has exactly two factors.
• 1 is neither prime nor composite.
• 2 is the smallest and only even prime.
• There are 25 primes between 1 and 100.
• There are infinitely many primes (proved by Euclid).
• Consecutive numbers are always co-prime.
• Prime factorisation of a number is unique.
• A number is divisible by 4 if its last two digits form a multiple of 4.
• A number is divisible by 9 if its digit sum is a multiple of 9.
• Divisible by 6 means divisible by both 2 and 3.
• The LCM of 3 and 5 is 15 — the first "idli-vada".

13. Exam-pattern Questions with Detailed Answers

14. Quick Revision Summary