Year 7 Number properties and patterns: free practice test with answers

Sample number properties and patterns questions with answers

Try each one before you open the answer.

1. Factors and multiples

   List all the factors of $36$.

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   Key idea: Find factor pairs that multiply to give the number, stopping when the pairs start to repeat.

   1. $1 \times 36$, $2 \times 18$, $3 \times 12$, $4 \times 9$, $6 \times 6$
   2. $5$ does not divide $36$, and after $6$ the pairs repeat
   3. List them in order: 1, 2, 3, 4, 6, 9, 12, 18, 36

   Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36

2. Highest common factor and lowest common multiple

   Find:

   (a) the highest common factor (HCF) of $42$ and $70$

   (b) the lowest common multiple (LCM) of $6$ and $8$

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   Key idea: The HCF is the largest number that divides both numbers. The LCM is the smallest number that both numbers divide into.

   1. (a) Factors of $42$: 1, 2, 3, 6, 7, 14, 21, 42
   2. (a) Factors of $70$: 1, 2, 5, 7, 10, 14, 35, 70
   3. (a) The largest common factor is $14$
   4. (b) Multiples of $6$: $6, 12, 18, 24, \ldots$; multiples of $8$: $8, 16, 24, \ldots$
   5. (b) The first common multiple is $24$

   Answer: (a) $14$ (b) $24$

3. Divisibility tests

   Use divisibility tests to decide whether $4572$ is divisible by:

   (a) $3$

   (b) $9$

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   Key idea: A number is divisible by $3$ if its digit sum is divisible by $3$, and by $9$ if its digit sum is divisible by $9$.

   1. Digit sum: $4 + 5 + 7 + 2 = 18$
   2. (a) $18$ is divisible by $3$, so $4572$ is divisible by $3$
   3. (b) $18$ is divisible by $9$, so $4572$ is divisible by $9$ (in fact $4572 = 9 \times 508$)

   Answer: (a) Yes (b) Yes

4. Using indices

   Evaluate $4^{3}$.

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   Key idea: $4^3$ means $4 \times 4 \times 4$.

   1. $4 \times 4 = 16$
   2. $16 \times 4 = 64$

   Answer: $64$

5. Prime decomposition

   Express $72$ as a product of prime factors using exponent notation.

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   Key idea: A factor tree splits a number into factor pairs, working down until every branch ends in a prime number. Collect repeated primes using exponent notation.

   1. Split $72$ into $8 \times 9$.
   2. Split $8$ into $2 \times 4$, then $4$ into $2 \times 2$. So $8 = 2 \times 2 \times 2$.
   3. Split $9$ into $3 \times 3$.
   4. Collect all prime factors: 2, 2, 2, 3, 3.
   5. Write using exponents:\

   $$72 = 2^3 \times 3^2$$

   Answer: $72 = 2^3 \times 3^2$

6. Squares and square roots

   Find the value of $8^2$.

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   Key idea: $8^2$ means $8 \times 8$, not $8 \times 2$.

   1. Write the meaning of the notation: $8^2 = 8 \times 8$
   2. Multiply: $8 \times 8 = 64$

   Answer: $8^2 = 64$

7. Spatial patterns

   Matchsticks are used to make a row of squares joined side by side. One square uses $4$ matchsticks, two squares in a row use $7$ matchsticks, and three squares in a row use $10$ matchsticks.

   (a) Copy and complete the table for $1$ to $5$ squares (number of squares, number of matchsticks).

   (b) Write a rule for the number of matchsticks $m$ needed for $s$ squares.

   (c) How many matchsticks are needed for $25$ squares?

   (d) What is the largest number of squares that can be made with $100$ matchsticks?

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   Key idea: Each new square shares one side with the previous square, so it adds only $3$ matchsticks.

   1. (a) Add $3$ each time: $1 \to 4$, $2 \to 7$, $3 \to 10$, $4 \to 13$, $5 \to 16$
   2. (b) The first square needs $1$ extra matchstick on the left and every square needs $3$ more, so $m = 3s + 1$. Check: $s = 3$ gives $3 \times 3 + 1 = 10$ ✓
   3. (c) $m = 3 \times 25 + 1 = 76$
   4. (d) We need $3s + 1 \le 100$, so $3s \le 99$ and $s \le 33$
   5. (d) $33$ squares use $3 \times 33 + 1 = 100$ matchsticks exactly

   Answer: (a) 4, 7, 10, 13, 16 (b) $m = 3s + 1$ (c) $76$ (d) $33$ squares

8. Tables and rules

   A function has the rule $y = 3x + 2$. Complete the table of values for $x = 0, 1, 2, 3$.

   $x$	0	1	2	3
   $y$

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   Key idea: Substitute each $x$ value into the rule to find the matching $y$ value.

   1. $x = 0$: $y = 3(0) + 2 = 2$
   2. $x = 1$: $y = 3(1) + 2 = 5$
   3. $x = 2$: $y = 3(2) + 2 = 8$
   4. $x = 3$: $y = 3(3) + 2 = 11$

   $x$	0	1	2	3
   $y$	2	5	8	11

   Answer: $y$ values are 2, 5, 8, 11.

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 7 powers and prime factors worksheet
• Year 7 squares and square roots worksheet
• Year 7 tables of values and plotting points worksheet