Seahorse Series · Quiz 7 · Factors, Multiples, and Primes

The Guide

Before you begin

Numbers have families. This quiz is about two questions numbers ask about each other: which numbers build me? (my factors) and which numbers do I build? (my multiples). Factors look inward and run out quickly; multiples run outward and never stop. Confusing the two is the most common trouble on this topic, so the words earn their keep here. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.

Factors are the rectangles you can build

If you did Quizzes 4 and 5, you have met the rectangle already: it showed multiplication, then found a missing side. Now it answers a third question. A factor of a number is any whole number that divides it evenly, with nothing left over. And every factor pair is a rectangle you could actually build with that many tiles:

Drawn to scale. Twelve tiles arrange three ways; thirteen tiles arrange only one. A prime is a number that cannot be rearranged.

Finding every factor: the pair walk

Factors arrive two at a time, so hunt them in pairs. Start at 1 and walk upward, testing each number: does it divide evenly? If yes, write down the pair. Stop when the two ends of your list meet. For 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6, and then 5 fails while 6 is already written, so the walk is over. Eight factors, found in four steps.

Multiples run the other way

A multiple of a number is what you land on when you skip-count by it: the multiples of 7 are 7, 14, 21, 28, 35, and onward forever. Every number is a multiple of itself, and no list of multiples ever ends. The plainest way to hold the difference: 6 is a factor of 24, and 24 is a multiple of 6. One relationship, read from its two ends.

Prime and composite

A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. A composite number has more than two, so it can be arranged more than one way. And 1 is neither, because it has only one factor. That isn't a technicality invented to annoy anyone: primes are the building blocks, and 1 builds nothing, since multiplying by it changes nothing at all.

The one prime that surprises people: 2 is prime, and it is the only even prime. Every other even number has 2 as a factor and so has at least three. Being even is not what makes a number composite; having more than two factors is.

The divisibility tests

These are the quick checks that save you from long division. They pay for themselves for the rest of the series, especially when fractions arrive and need simplifying:

Divides by	The test	Example
2	The last digit is even (0, 2, 4, 6, 8)	3,474 passes
3	Add the digits; if that sum divides by 3, so does the number	471 → 4+7+1 = 12, and 12 divides by 3
4	The last two digits form a number divisible by 4	5,116 → 16 divides by 4
5	The last digit is 0 or 5	2,895 passes
6	It must pass both the 2 test and the 3 test	342 is even, and 3+4+2 = 9
9	Add the digits; if that sum divides by 9, so does the number	6,183 → 6+1+8+3 = 18
10	The last digit is 0	4,320 passes

The table is wider than your screen: slide it sideways with your finger, or turn your phone.

The digit-sum tests for 3 and 9 look like magic and aren't. Every 10 is a 9 plus 1, every 100 is a 99 plus 1, and 9 and 99 already divide by 3 and by 9. So the only part of a number that can leave anything over is those extra 1s: one for each hundred, one for each ten, plus the ones. Added up, those are the digits themselves.

Prime factorization: a number's recipe

Break a number into factors, then break those, until only primes remain. That is its prime factorization, and here is the remarkable part: every whole number has exactly one, no matter which way you start breaking it. 60 becomes 2 × 2 × 3 × 5, written 2² × 3 × 5, whether you begin with 6 × 10 or 4 × 15. The recipe is the number's fingerprint, and it does real work later: simplifying fractions, finding common denominators, taking square roots.

Worked Examples

Three to study before you start

Example 1 · The pair walk

All factors of 24. Walk upward: 1 × 24, 2 × 12, 3 × 8, 4 × 6. Test 5, which fails. Test 6, already on the list, so the ends have met and the walk is done. Factors: 1, 2, 3, 4, 6, 8, 12, 24. Eight factors mean eight rectangles, counting each on its side.

Example 2 · Running the tests

Which of 2, 3, 4, 5, 6, 9, 10 divide 5,472? Last digit 2, so it passes 2. Digits add to 5+4+7+2 = 18, which divides by both 3 and 9, so it passes 3 and 9. Last two digits are 72, which divides by 4. It passes 2 and 3, so it passes 6. It ends in neither 0 nor 5, so 5 and 10 both fail. Answer: 2, 3, 4, 6, 9. Seven questions, no long division, about fifteen seconds.

Example 3 · Breaking down to the recipe

Prime factorization of 60. Start anywhere: 60 = 6 × 10. Neither is prime, so break both: 6 = 2 × 3, and 10 = 2 × 5. Now everything is prime: 2 × 2 × 3 × 5, or 2² × 3 × 5. Start instead from 60 = 4 × 15 and you arrive at the same four primes. The recipe doesn't care how you began.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Write your answers on the lines. Questions 3 and 7 are multiple choice: choose the one best answer.

List all the factors of 18.
List the first five multiples of 7.
Which statement is true?
- A) 6 is a multiple of 24, and 24 is a factor of 6.
- B) 6 is a factor of 24, and 24 is a multiple of 6.
- C) Both 6 and 24 are factors of each other.
- D) Neither is a factor or multiple of the other.
Is 51 prime? Answer yes or no, and give your reason.
Which of 2, 3, 4, 5, 6, 9, 10 divide evenly into 2,754? Use the tests, not long division.
Write the prime factorization of 60.

▶ VideoA video rest stop, for anyone who has kept going this far.

Take two minutes with these seahorses.

Which of these numbers is prime?
- A) 51
- B) 57
- C) 61
- D) 91
Is 1 a prime number? Answer and explain in one sentence.
A hall holds 48 chairs, and they must be set in equal rows with none left over. How many different row counts are possible?
Two maintenance crews start work at the same building today. One returns every 6 days, the other every 8 days. In how many days will both be at the building again on the same day?

Answer Key

Check your work

Open the key — after you've finished all ten

1 1, 2, 3, 6, 9, 18

The pair walk: 1 and 18, 2 and 9, 3 and 6. Test 4, which fails; test 5, which fails; 6 is already written, so the ends have met. Six factors, six rectangles. The most common slip is stopping before the walk is finished: keep going until your two ends meet.

2 7, 14, 21, 28, 35

Skip-counting by 7. Note that the list starts at 7 itself, not at 14: every number is its own first multiple. And this list has no end, which is the deepest difference between multiples and factors.

3 B.

The small number is the factor, the big number is the multiple: 6 builds 24, and 24 is built. A) reverses the pair, which is the single most common error on this topic. C) confuses the two words. D) denies the relationship entirely. Say it out loud until it sticks: the factor is the ingredient, the multiple is the dish.

4 No: 51 = 3 × 17.

51 looks prime because it's odd and ends in 1, but odd is not the test. Run the digit sum: 5 + 1 = 6, which divides by 3, so 3 divides 51. Any reason naming a third factor earns the credit. This is the classic false prime, and the divisibility tests are exactly what catches it.

5 2, 3, 6, and 9

Last digit 4, so 2 passes. Digits add to 2+7+5+4 = 18, so 3 and 9 both pass. It passes 2 and 3, so 6 passes. Last two digits are 54, which does not divide by 4, so 4 fails. It ends in neither 0 nor 5, so 5 and 10 fail. If you claimed 4, remember the test looks only at the last two digits — and 54 is not a multiple of 4.

6 2 × 2 × 3 × 5, or 2² × 3 × 5

Break by any route you like; every route lands here. If your answer still contains a composite number, such as 4 × 15 or 2 × 30, the breaking isn't finished: keep going until nothing but primes remain.

7 C: 61.

Every wrong answer is a false prime, and each is caught by a test. A) 51 = 3 × 17, caught by the digit sum, 5+1 = 6. B) 57 = 3 × 19, caught the same way, 5+7 = 12. D) 91 = 7 × 13, which no digit test catches — for that one you simply try the small primes in turn, and 7 works. 61 survives every trial: 2, 3, 5, and 7 all fail. That is far enough, because 8 × 8 = 64 is already bigger than 61: any factor larger than 8 would need a partner smaller than 8, and every one of those has been tried.

8 No. 1 has only one factor, and a prime must have exactly two.

1 is neither prime nor composite. The rule isn't arbitrary bookkeeping: primes are the building blocks that every number is made from, and 1 builds nothing, since multiplying by it leaves a number exactly as it was. Let 1 count as prime and every recipe in question 6 would have infinitely many versions.

9 10 different row counts

This question is asking for factors, though it talks about chairs. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 — ten of them, so ten arrangements, from one long row of 48 to 48 rows of one. Every seating chart that comes out even is a rectangle, and every rectangle is a factor pair.

10 24 days

And this one is asking for multiples. Crew one returns on days 6, 12, 18, 24; crew two on days 8, 16, 24. The first day both lists share is 24 — the least common multiple. Hold questions 9 and 10 side by side: same topic, opposite directions. Factors ask what divides a number; multiples ask what a number divides. Calendars and schedules are multiples problems; arrangements and equal shares are factors problems.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2	Listing factors and multiples	The pair walk until the ends meet; skip-counting that starts at the number itself.
3, 9, 10	Telling the two apart	The factor is the ingredient, the multiple is the dish. Arrangements are factors; schedules are multiples.
4, 7, 8	Primes, and the false primes	Exactly two factors. Odd is not the test; run the divisibility tests, then try the small primes in turn.
5	The divisibility tests	Reread the table; the 4 test reads only the last two digits, and the 6 test needs both the 2 test and the 3 test.
6	Prime factorization	Keep breaking until nothing composite is left; the recipe is the same by every route.

Eight or more right: the family trees are yours, and fractions will thank you — on to Quiz 8. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. Build rectangles from coins or beans with your teacher, 12 of them and then 13, and see for yourself which numbers refuse to rearrange. Then come back to these same ten.

The Company · An Interlude

Fellow travelers

A change of watch at the masthead: a new supporter stands to the left today, in silver. Below, that same creature seen whole, and a very small relation who has made a career of not being found.

A tiny orange pygmy seahorse gripping a branch of red gorgonian coral, its body covered in matching orange bumps — A pygmy seahorse (*Hippocampus bargibanti*) on its gorgonian coral, photographed by Steve Rosenberg. Barely the size of a fingernail, and covered in bumps that copy the coral's own. It was discovered only because someone examined the coral after it had already been collected.

A seahorse in silver and black on a dark field, spines and body rings catching the light — Today's new supporter, seen whole. Color turned off again, and again the structure carries it: rings, spines, the curl of the tail. A number has a shape like this too, and Quiz 7 is about reading it.

Prints only this gallery, full size. The main print button stays photograph-free to spare your ink.