The Lumenary · Grades 3–5

Factors, multiples and primes

Name: ______________________

Date: ______________________

You will learn to

List the factors of numbers up to $100$ in pairs, and find common factors and common multiples
Tell factors from multiples
Classify numbers as prime or composite (and know that $1$ is neither), and use the sieve of Eratosthenes
Write a whole number as a product of primes with a factor tree
Explain the divisibility tests for $2$ , $3$ , $5$ , $9$ and $10$

1.
List all the factors of $36$ , separated by commas.
2.
Which sentence is true?
- A. $18$ is a factor of $6$
- B. $6$ is a multiple of $18$
- C. $6$ is a factor of $18$ , and $18$ is a multiple of $6$
- D. $6$ is both a factor and a multiple of $18$
3.
Is the number $1$ prime, composite, or neither?
- A.Prime
- B.Composite
- C.Neither
4.
Which of these numbers are prime? (Choose all that apply.)
- A. $2$
- B. $9$
- C. $29$
- D. $51$
- E. $91$
- F. $97$
5.
What is the greatest common factor of $24$ and $36$ ?
6.
In Nairobi, one bus leaves the station every $4$ minutes and another every $6$ minutes. Both leave together right now. After how many minutes do they next leave together?
7.
Which of these writes $84$ as a product of primes?
- A. $2 \times 2 \times 3 \times 7$
- B. $4 \times 3 \times 7$
- C. $2 \times 42$
- D. $2 \times 3 \times 14$
8.
Use the digit-sum test. Which of these numbers are divisible by $9$ ? (Choose all that apply.)
- A. $738$
- B. $1234$
- C. $9909$
- D. $5551$
9.
Lean writes b ∣ a to mean " $b$ is a factor of $a$ ", that is, $a = b \times k$ for some $k$ . Which Lean statement says that $3$ is a factor of $12$ ?
- A.3 ∣ 12
- B.12 ∣ 3
- C.12 % 3 = 3
- D.3 * 12 = 36
10.
Write is_prime(n) that returns True if n is prime and False otherwise. Try every possible factor d from 2 upwards, but stop as soon as d * d > n (the lesson explains why that is enough). The starter uses the mistaken idea that odd numbers are prime, so the tests fail.
Computational — Practice

Answer key — Factors, multiples and primes

1.
The factor pairs are $1 \times 36$ , $2 \times 18$ , $3 \times 12$ , $4 \times 9$ and $6 \times 6$ . So the factors are $1, 2, 3, 4, 6, 9, 12, 18, 36$ .
2.
$18 = 6 \times 3$ , so $6$ is a factor of $18$ and $18$ is a multiple of $6$ . Factors are the small pieces; multiples are built from them.
3.
$1$ is neither. Both 'prime' and 'composite' are defined only for numbers greater than $1$ . If $1$ counted as prime, $6 = 2 \times 3 = 1 \times 2 \times 3$ would have more than one prime factorisation.
4.
$2$ , $29$ and $97$ are prime. $9 = 3 \times 3$ , $51 = 3 \times 17$ and $91 = 7 \times 13$ are composite. For a number below $100$ , if none of $2, 3, 5, 7$ divides it (and it is bigger than $1$ ), it is prime.
5.
The common factors are $1, 2, 3, 4, 6, 12$ , so the greatest common factor is $12$ . Check: $24 = 12 \times 2$ and $36 = 12 \times 3$ .
6.
The first bus leaves at $4, 8, 12, \dots$ and the second at $6, 12, 18, \dots$ minutes. The first time on both lists is $12$ : the least common multiple of $4$ and $6$ .
7.
$84 = 2 \times 42 = 2 \times 2 \times 21 = 2 \times 2 \times 3 \times 7$ . Every factor is prime, so the tree stops here. Any other tree gives the same primes.
8.
Digit sums: $738 \to 18$ , $1234 \to 10$ , $9909 \to 27$ , $5551 \to 16$ . So $738$ and $9909$ are divisible by $9$ , and $1234$ and $5551$ are not.
9.
3 ∣ 12. Read b ∣ a as " $b$ divides $a$ ": the factor is written first. The Lean theorem three_and_twelve proves 3 ∣ 12 and also that 12 ∣ 3 is false.
10.
```
def is_prime(n):
    if n < 2:
        return False
    d = 2
    while d * d <= n:
        if n % d == 0:
            return False
        d = d + 1
    return True
```
Why stop at d * d > n? If $n = a \times b$ with both factors bigger than $1$ , they can't both be bigger than the square root, because then $a \times b$ would be bigger than $n$ . So the smaller factor would already have been found.