seegongsik
Grade 5 (age 10-11)

grade 5 Primes and Composites

A prime number has exactly two divisors, 1 and itself; a composite number has other divisors as well. 2, 3, 5 and 7 are primes, while numbers such as 4 = 2 × 2 and 6 = 2 × 3, which can be written as a product of smaller numbers, are composite. 1 has only one divisor, so it is neither prime nor composite. The sieve of Eratosthenes is an ancient method that crosses out the multiples of 2, then of 3 and so on, leaving only the primes. In the first picture, choose how far to cross out multiples and sift the numbers 1 to 50; in the second, pick a number and decide from its divisors whether it is prime or composite.

What is a prime?

⭐ Numbers that only 1 and themselves divide
①A number with exactly two divisors, 1 and itself, is a prime
②2, 3, 5, 7, 11, 13, 17, 19 and so on are primes, and 2 is the only even prime
③A number with 3 or more divisors is composite; it can be written as a product of smaller numbers, like 12 = 3 × 4
④1 has only one divisor, 1 itself, so it is neither prime nor composite

The sieve of Eratosthenes

3
🔢 Sifting through the sieve
①Cross out 1, keep 2, and cross out all the multiples of 2: 4, 6, 8 and so on
②Next keep 3 and cross out its multiples: 6, 9, 12 and so on. The picture starts at this stage, so composites such as 25 and 49 are still left
③Cross out the multiples of 5 and 7 as well and every composite up to 50 is gone, because 7 × 7 = 49 while the next prime, 11, gives 11 × 11 = 121, which is beyond 50
④The numbers left at the end are the primes, and the line under the picture changes from 'Left so far' to 'Primes'

Prime or composite?

🔎 Deciding by counting divisors
①Pick a number in the second picture and it lists all its divisors and counts them
②2 divisors means prime, 3 or more means composite. The starting number, 12, has 6 divisors, 1, 2, 3, 4, 6 and 12, so it is composite
③If any other divisor sits between the 1 and the number itself at the two ends, the number is composite
④Pick 1 and you are told it has only one divisor, so it is neither prime nor composite
12

Summary

Prime
Natural number whose divisors are only 1 and itself
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
Composite
Natural number with 3+ divisors
4, 6, 8, 9, 10, 12, 14, 15, ...
1 is special
1 is neither prime nor composite
Because its only divisor is 1
🎯 At a glance
①Prime: exactly two divisors, 1 and itself
②Composite: 3 or more divisors
③1 is neither prime nor composite
④The smallest prime is 2, and it is the only even prime
⑤Crossing out multiples with the sieve of Eratosthenes leaves only the primes

Examples and Unit Test

Example 1
Is 7 prime or composite?
1
The divisors of 7 are just 1 and 7.
2
With only two divisors, 1 and itself, 7 is prime.
▸ Prime
Pick 7 in the second picture and only the two end boxes appear.
Example 2
Is 12 prime or composite?
1
The divisors of 12 are 1, 2, 3, 4, 6 and 12: six in all.
2
It has 3 or more divisors, so 12 is composite.
▸ Composite
12 is the starting number in the second picture, with 2, 3, 4 and 6 in green between 1 and 12.
Unit Test
Which of these is not a prime?
①2
②3
③9
④11
▸ ③ 9
1
The divisors of 9 are 1, 3 and 9: three of them.
2
With three divisors, 9 is composite, while 2, 3 and 11 each have two divisors and are prime.
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Divisors & Multiples
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Reduction & Common Denominator

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