Prime numbers to 100: the list of all 25, why 1 is not prime and 2 is, how to check whether any number is prime in four steps, the Sieve of Eratosthenes, prime factors and factor trees, the primes to 200 and how many there are to 1,000, and a printable chart with practice

"Prime numbers to 100" is typed in the UK about 2,400 times a month, with "all prime numbers" another 3,200 and "prime numbers list" 1,800, and in the US "prime numbers list" 14,000 times; most of the typists are Year 5 and 6 children (or Grade 4 and 5) with a homework sheet asking them to circle the primes on a hundred square. This page lists the 25, explains the definition and the two questions every child asks (why not 1, why 2), gives a four-step check that works for any number, shows the Sieve of Eratosthenes, goes on to prime factors, extends the list to 200 and counts the primes to 1,000, and ends with a printable chart and practice set. Each prime to 100 also has its own page in our maths library, with the check written out.

Download the prime numbers chart with practice (PDF) (A4, the hundred square with primes marked and the list to 1,000 on page 1, the sieve and the practice set with answers on page 2, free).

The 25 prime numbers to 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.

By ten:

England's Year 5 objective is to "establish whether a number up to 100 is prime and recall prime numbers up to 19": the first eight (2, 3, 5, 7, 11, 13, 17, 19) are to be known by heart, the rest to be worked out.

What a prime number is

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. 7 is prime because the only ways to make it by multiplying whole numbers are 1 × 7 and 7 × 1. 8 is not, because 2 × 4 works too, so it has four factors (1, 2, 4, 8). A number with more than two factors is composite.

• 1 is not prime. It has one factor, not two. It is not composite either; it is the unit, in a class of its own. (The deeper reason: every number must break into primes in exactly one way, and that only works if 1 is left out.) The is 1 a prime number page has the full argument.
• 2 is prime, and the only even one, because every other even number has 2 as an extra factor. "All primes are odd" is wrong by exactly one number.
• 0 is not prime: it has infinitely many factors (every number divides 0).
• Primes are the building blocks: every whole number above 1 is either prime or a product of primes, in one way only (the fundamental theorem of arithmetic). 60 = 2 × 2 × 3 × 5, and there is no other set of primes that multiplies to 60.

How to check whether a number is prime

1. Is it even? If it ends in 0, 2, 4, 6 or 8 and is not 2 itself, it is not prime.
2. Do its digits add to a multiple of 3? Then 3 divides it (51: 5 + 1 = 6, so 51 = 3 × 17, not prime). If it is not 3 itself, it is out.
3. Does it end in 5 or 0? Then 5 divides it (unless it is 5).
4. Try 7, then 11, 13 and the next primes, but stop when the prime you are testing squared is bigger than the number. If nothing has divided it by then, it is prime.

The stopping rule is what makes the check short. A composite number must have a factor no bigger than its square root (if both factors were bigger than the root, their product would be bigger than the number). So:

• Up to 100 (√100 = 10): test 2, 3, 5, 7 only.
• Up to 200 (√200 ≈ 14.1): test 2, 3, 5, 7, 11, 13.
• Up to 1,000 (√1000 ≈ 31.6): test the primes up to 31.

Worked: is 91 prime? Odd; 9 + 1 = 10, not a multiple of 3; does not end in 5; 91 ÷ 7 = 13. Not prime. Is 97 prime? Odd; 9 + 7 = 16; not 5; 97 ÷ 7 = 13 remainder 6. Nothing left to test (11² = 121 > 97). Prime. Is 221 prime? Not 2, 3 or 5; 221 ÷ 7 = 31 r 4; 221 ÷ 11 = 20 r 1; 221 ÷ 13 = 17. Not prime. The divisibility rules page has the tests for 4, 6, 8, 9 and 11 as well.

The Sieve of Eratosthenes

The Greek mathematician Eratosthenes (about 240 BC) found the primes without dividing anything, by crossing out. On a hundred square:

1. Cross out 1.
2. Circle 2. Cross out every second number after it (4, 6, 8, ...).
3. Circle 3, the next number not crossed out. Cross out every third number (6, 9, 12, ...; some are already gone).
4. Circle 5. Cross out every fifth number.
5. Circle 7. Cross out every seventh number (the first new one is 49, since 14, 21, 28, 35 and 42 are already out).
6. Stop. Everything not crossed out is prime, because the next candidate, 11, has 11² = 121, off the square.

What is left is the 25 primes, in about two minutes of colouring. It is the single best homework activity for the topic, because the child sees why the multiples of a prime are not prime and why the process stops at 7. The PDF has a blank hundred square for it.

Prime factors and factor trees

Every composite number splits into primes. A factor tree does it by repeated splitting: 72 → 8 × 9 → (2 × 4) × (3 × 3) → 2 × 2 × 2 × 3 × 3 = 2³ × 3². Any route gives the same primes. Prime factorisation is the tool behind highest common factor (HCF) and lowest common multiple (LCM) in Year 6 and 7:

• 72 = 2³ × 3², 60 = 2² × 3 × 5. HCF = 2² × 3 = 12 (the primes they share, lowest powers). LCM = 2³ × 3² × 5 = 360 (every prime, highest powers).

It is also why primes matter outside the classroom: the security of online banking rests on the fact that multiplying two 300-digit primes is easy but recovering them from the product is, with present computers, impossibly slow.

The primes to 200

The 21 primes between 100 and 200: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199. That makes 46 primes up to 200.

Patterns worth noticing with a child: every prime after 3 ends in 1, 3, 7 or 9 (never 0, 2, 4, 5, 6 or 8). Every prime after 3 is one more or one less than a multiple of 6 (5 and 7, 11 and 13, 17 and 19, 23 and 25 (not prime), 29 and 31). Twin primes are pairs two apart: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199). Whether there are infinitely many twin primes is still unproven.

How many primes there are to 1,000

168 primes to 1,000, 303 to 2,000, 1,229 to 10,000. They thin out but never run out: Euclid's proof is three lines (multiply any list of primes together, add 1, and the result is divisible by none of them, so a prime is missing from the list). The largest prime known in 2024 had over 41 million digits. The full list to 1,000 is on the PDF.

Myths that cost marks

• "1 is prime." No: one factor only.
• "All odd numbers are prime." 9, 15, 21, 25, 27 are odd and composite.
• "All primes are odd." 2.
• "Big numbers ending in 7 are prime." 27, 57, 77, 87 are not.
• "91 is prime." The classic trap: 7 × 13. So is 51 (3 × 17), 57 (3 × 19) and 87 (3 × 29).
• "A prime factor can be 1." Factor trees stop at primes; 1 never appears.

Where it sits in each curriculum

Practice (answers in the PDF)

1. Write all the prime numbers between 20 and 50.
2. Is 87 prime? Show the check.
3. Is 101 prime? Which primes do you need to test, and why can you stop there?
4. Write 84 as a product of its prime factors.
5. Find the HCF and the LCM of 36 and 90 using prime factors.
6. Which numbers between 1 and 30 have exactly three factors? (Hint: they are squares of primes.)

Download the chart and the practice set (PDF)

Where to get help

A 1-on-1 online maths class with a tutor who holds a master's in mathematics and teaches KS2 and KS3, the Australian Years 5 to 8, CBSE Class 5 to 8 and US Grades 4 to 7 every week: primes taught by the sieve and the square-root rule, not by a list to memorise, with factor trees, HCF and LCM following in the same term. The maths library has a page for every prime question children type; the free 30-minute first class is a real lesson on this week's homework. The partner page is square numbers 1 to 100 and the cube numbers.

Questions parents ask

See how we teach this, 1-on-1 online →