Factors

14 Grade 7 Mathematics questions on Factors, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   2 marks

   Three buckets have capacities of 16 litres, 12 litres and 20 litres. The buckets were filled with water using a container. What is the capacity of the largest container that fills each bucket in an exact number of times?

   A. 240 litres

   B. 4 litres

   C. 48 litres

   D. 60 litres

   Marking scheme

   B. $4$ litres — 'fills each exactly' means the container's size must divide all three, and 'largest' means the highest common factor of $16$, $12$ and $20$, which is $4$. $8$ and $12$ each divide one of them but not all three, and $2$ divides all three but is not the largest.

2. Question 2

   3 marks

   Find all the prime numbers between 30 and 50.

   (3 marks)

   Marking scheme

   A prime number has exactly two factors, $1$ and itself. Test each number from $31$ to $49$ for division by $2$, $3$, $5$ and $7$. (1)

   The even numbers and the multiples of $3$, $5$ and $7$ are all removed, leaving $31$, $37$, $41$, $43$ and $47$. (1)

   Checking one: $47$ is not divisible by $2$, $3$, $5$ or $7$, and $7^2=49$ is already past $47$, so no larger factor is possible. (1)

   Final answer: $31$, $37$, $41$, $43$ and $47$.

3. Question 3

   2 marks

   Three pieces of wire are 42 cm, 56 cm, and 70 cm long. What is the greatest length that can be cut from each piece without a remainder?

   (2 marks)

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4. Question 4

   1 mark

   What is the LCM of 8 and 12?

   A. 12

   B. 24

   C. 96

   D. 48

   (1 mark)

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5. Question 5

   3 marks

   The GCD of two numbers is 12 and their LCM is 180. If one number is 36, find the other.

   (3 marks)

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6. Question 6

   5 marks

   A school bought some rice to be used for 6 days. On the first day, the school used $\frac{1}{6}$ of the rice. On the second day, $\frac{1}{5}$ of the remaining rice was used. The rest was shared equally for the last 4 days. On the last day the school used 37.5 kg.

   Determine:

   (a) the fraction of rice shared equally for the last 4 days. (2 marks)

   (b) the mass of rice that the school had bought. (3 marks)

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7. Question 7

   1 mark

   How many factors does 60 have?

   A. 12

   B. 10

   C. 6

   D. 8

   (1 mark)

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8. Question 8

   5 marks

   A school bought some rice to be used for 6 days. On the first day, the school used $\frac{1}{6}$ of the rice. On the second day, $\frac{1}{5}$ of the remaining rice was used. The rest was shared equally for the last 4 days. On the last day the school used 37.5 kg.

   Determine:

   (a) the fraction of rice shared equally for the last 4 days. (2 marks)

   (b) the mass of rice that the school had bought. (3 marks)

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9. Question 9

   3 marks

   Express 1,260 as a product of prime factors using a factor tree.

   (3 marks)

   Marking scheme

   Divide by the smallest prime that goes in, again and again. $1,260÷2=630$, and $630÷2=315$. (1)

   $315$ is odd, so move to $3$: $315÷3=105$, and $105÷3=35$. (1)

   $35=5\times 7$, and both are prime, so the tree stops. (1)

   Final answer: $1,260=2^2\times 3^2\times 5\times 7$.

10. Question 10

    2 marks

    Three buckets have capacities of 16 litres, 12 litres and 20 litres. The buckets were filled with water using a container. What is the capacity of the largest container that fills each bucket in an exact number of times?

    A. 240 litres

    B. 4 litres

    C. 48 litres

    D. 60 litres

    Marking scheme

    B. $4$ litres — 'fills each exactly' means the container's size must divide all three, and 'largest' means the highest common factor of $16$, $12$ and $20$, which is $4$. $8$ and $12$ each divide one of them but not all three, and $2$ divides all three but is not the largest.

11. Question 11

    1 mark

    Which is a prime number?

    A. 33

    B. 39

    C. 37

    D. 51

    (1 mark)

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12. Question 12

    1 mark

    Three buckets hold 16, 12, and 20 litres. What is the largest container that fills each exactly?

    A. 4 litres

    B. 12 litres

    C. 8 litres

    D. 2 litres

    (1 mark)

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13. Question 13

    1 mark

    Express 300 as a product of prime factors.

    A. 2² × 3² × 5

    B. 2² × 3 × 5²

    C. 2 × 3 × 5³

    D. 2³ × 3 × 5

    (1 mark)

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14. Question 14

    1 mark

    What is the GCD of 36 and 54?

    A. 18

    B. 9

    C. 6

    D. 36

    (1 mark)

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