Secondary 1 Divisibility, Factors and Multiples Worksheet

All 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Divisibility Rules

   For each number below, say which of $2$, $3$, $4$, $5$, $6$, $9$ and $10$ divide it. Name the rule you used each time — do not do the divisions.

   1. $1260$
   2. $4095$
   3. $2736$
2. Q2Divisors, Multiples and Factorization

   Answer each part and show the pairing or the multiplication you used.

   1. List all the divisors of $84$.
   2. List the first six multiples of $14$, starting at $14$.
   3. Is $84$ a multiple of $14$? Is $84$ a divisor of $168$? Justify both answers with one multiplication each.
3. Q3Exponential Notation

   Use exponential notation carefully.

   1. Write $7\times 7\times 7\times 7$ as a power, then give its value.
   2. Write $2·2·2·5·5$ as a product of two powers, then give its value.
   3. In $6^3$, name the base and name the exponent, and read the power aloud in words.
   4. Evaluate $3^4$ and $5^3$.
4. Q4Exponents –

   Evaluate, respecting the order of operations.

   1. $4^3$
   2. ${10}^4$
   3. $1^{12}$
   4. $6^2-5^2$
   5. $2+3\times 4^2$
5. Q5Factoring a Number

   Build a factor tree for each number, then write its prime factorization in exponential notation, with the prime factors in increasing order.

   1. $360$
   2. $588$
   3. $1000$
6. Q6Methods for Finding the LCM and the GCF Simultaneously

   For each pair, divide both numbers side by side by a common prime factor, again and again, until the two numbers left share no common factor. Then read off the GCF and the LCM from your work.

   1. $84$ and $126$
   2. $60$ and $96$
7. Q7Square and Cubic Numbers –

   Answer using the meaning of a square number and of a cubic number.

   1. List every square number from $100$ to $300$.
   2. List every cubic number from $1$ to $500$.
   3. Find a number less than $100$, other than $1$, that is both a square number and a cubic number, and show both ways of building it.
8. Q8The Greatest Common Factor (GCF)

   For a science fair, Amélie has $90$ blue LEDs and $126$ red LEDs. She wants to build identical kits, using every LED and leaving none over.

   1. Write $90$ and $126$ as products of prime factors, and use them to find the GCF.
   2. What is the greatest number of kits she can build?
   3. How many blue and how many red LEDs go into each kit?
9. Q9The Least Common Multiple (LCM)

   Two buses leave the Longueuil terminus together at $6$:$00$. The $51$ leaves every $18$ minutes and the $24$ leaves every $24$ minutes, all morning.

   1. Find the LCM of $18$ and $24$ using prime factorization.
   2. At what time do the two buses next leave together?
   3. List every time before $11$:$00$ at which they leave together.
10. Q10The Properties of Natural Numbers

    Consider $1$, $2$, $27$, $41$, $49$, $51$, $64$ and $91$.

    1. Sort them into prime, composite, and neither. For each composite number, give one factor pair.
    2. Which of them are square numbers?
    3. Which of them are odd?
11. Q11The Root of a Number \textbar{} Secondary 1 and 2

    Evaluate without a calculator, and show the perfect square you recognized.

    1. $\sqrt{169}$
    2. $\sqrt{2500}$
    3. $\sqrt{36}+\sqrt{64}$
    4. $\sqrt{9\times 16}$
    5. A square vegetable garden has an area of $225{\text{m}}^2$. Find the length of one side, then the length of fence needed to go around it.
12. Q12Synthesis — drawing on several sheets in this topic

    A rectangular concrete slab at a skatepark measures $168\text{cm}$ by $252\text{cm}$. It is to be covered exactly by identical square tiles, laid in full rows and columns, with no tile cut.

    1. Write $168$ and $252$ as products of prime factors.
    2. What is the largest possible side length of a tile? Justify using the two factorizations.
    3. How many tiles of that size are needed? Check your count using the areas.

The 12 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The factorization is the answer sheet

Almost every question on this topic is easier from the prime factorization than from the number itself, and Q5 exists to build that habit before you need it. Once 360 has been written as 2³ × 3² × 5, questions about what divides it, what it shares with another number and what the two of them have in common are all answered by reading, not by dividing. Q2 and Q10 look like list questions; they are really questions about structure.

Divisibility rules: the instruction is not to divide (Q1)

Q1 asks which of 2, 3, 4, 5, 6, 9 and 10 divide each number, and tells you to name the rule rather than perform the division. That is the point of the question, so an answer with the right divisors and no rule named has answered half of it.

The rules fall into three families. Some read the last digit — even for 2, a 0 or 5 for 5, a 0 for 10. Some add the digits — a digit sum divisible by 3 means the number is, and the same test with 9. And one reads the last two digits: if they form a number divisible by 4, so is the whole thing. The rule for 6 is the only combined one — divisible by 2 and by 3 at the same time.

Divisors come in pairs, and the pairing tells you when to stop (Q2)

List divisors by pairing from the outside in: 1 × 84, 2 × 42, 3 × 28, 4 × 21, 6 × 14, 7 × 12. Each successful test hands you two divisors at once, and you can stop as soon as the two members of a pair meet or cross — anything larger has already appeared as the partner of something smaller. That is the difference between a complete list and a list with a hole in it.

Q2 then makes you use both words in the same breath. 84 is a multiple of 14 and 14 is a divisor of 84 — one multiplication, 84 = 14 × 6, justifies both statements. If you can produce the multiplication, you have the justification the question is asking for.

Exponents: an exponent belongs to its own base only (Q3, Q4)

A power is shorthand for repeated multiplication, and the two errors people make with it are both about scope. 2⁴ is 2 × 2 × 2 × 2 = 16, not 2 × 4. And in 5 × 2³ the exponent applies to the 2 alone, so the value is 5 × 8 = 40 — the 5 is not part of the base.

Q4 adds the order of operations: powers are evaluated before multiplication and addition, so 2 + 3 × 4² is 2 + 3 × 16 = 50. Naming the base and the exponent aloud, as Q3 asks, is a small ritual that prevents both mistakes, because it forces you to say exactly which number is being multiplied by itself.

Building a factor tree, and reading it afterwards (Q5)

Split the number into any two factors you like, keep splitting until every branch ends on a prime, then collect the primes in increasing order with exponents. The starting split does not matter — 360 as 36 × 10 and 360 as 4 × 90 reach the same 2³ × 3² × 5 — which is worth noticing, because that uniqueness is the reason a factorization can be trusted as an answer.

Always multiply back to check. It costs one line and it catches the branch you forgot to split.

GCF or LCM? Read the situation, not the numbers (Q8, Q9, Q12)

The arithmetic for these two is short; choosing between them is the actual question, and the word problems on this sheet are chosen to sit on opposite sides of that choice.

Q6 shows the method that produces both at once: write the two numbers side by side, divide both by a common prime, repeat until nothing is shared. The divisors you used multiply to the GCF; the GCF times the two leftovers gives the LCM. It is quicker than two separate hunts, and it makes visible what the two quantities have to do with each other.

Primes, squares and cubes (Q7, Q10)

Q10 sorts numbers into prime, composite and neither, and that third box exists for exactly one number: 1, which has only a single divisor and so is neither. The other trap is 2 — prime, and the only even prime, because every other even number already has 2 as a divisor.

Q7 asks for the square numbers in a range and the cubic numbers in another, which is just a matter of running through 10², 11², 12² and so on until you pass the ceiling. Its last part is more interesting: a number under 100 that is both a square and a cube. Sixty-four, as 8 × 8 and as 4 × 4 × 4 — and being able to show both constructions is the answer, not the number itself.

Roots of perfect squares (Q11)

A square root asks "what number, multiplied by itself, gives this?" — so it is the square question run backwards, and at this level the numbers are chosen to come out whole. Recognising 169 as 13², 2500 as 50² and 144 as 12² is the whole technique, and it is worth knowing the squares up to about 15 by heart.

Two habits from Q11. Work inside a root before taking it: for √(9 × 16), multiply to 144 first. And read the last part carefully — finding the side of a square garden from its area is one root, but the fence around it is four sides, so the root is a middle step rather than the answer.

Q12: the synthesis question

A rectangular slab covered exactly by identical square tiles, none cut. The tile's side has to divide both dimensions, so the largest one is the GCF — and the question asks you to justify that from the two factorizations rather than to assert it. Then count the tiles along each side and check the total against the areas. Two routes to the same count, again, is what makes the answer safe.

Getting the most out of it

Factor first, ask questions second

When a question involves two numbers and the words "common", "identical", "at the same time" or "with none left over", write both prime factorizations before anything else. The GCF and the LCM are then read off, and half the questions on this sheet stop being searches.

Name the rule, not just the answer

Several questions here ask which numbers divide, or why a claim fails, and they are marked on the reason. "Its digits add to 18, so 9 divides it" is an answer; a bare list of divisors is not. Practise writing the clause — it is the habit that later turns into a justification.

Check by multiplying back

Every factorization, every GCF and every LCM on this sheet can be verified in a single multiplication. Do it. It costs a line and it catches the missed branch, the forgotten divisor and the swapped GCF and LCM, which are between them nearly every error on the topic.

Common questions