Numbers and operations · Ages 12–13

H.C.F. and L.C.M. worksheet

Part of: Working with numbers

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Numbers and operations · Ages 12–13

Numsprite
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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.The highest common factor (H.C.F.) of 90 and 105 is .
Answer:
2.The lowest common multiple (L.C.M.) of 14 and 18 is .
Answer:
3.The highest common factor (H.C.F.) of 33 and 55 is .
Answer:
4.10 = 2¹ × 5¹ and 50 = 2¹ × 5². Their L.C.M. = 2^□ × 5^□
Fill in each blank
Answer:
5.432 = 2⁴ × 3³ and 72 = 2³ × 3². Their H.C.F. = 2^□ × 3^□
Fill in each blank
Answer:
6.144 = 2⁴ × 3² and 18 = 2¹ × 3². Their H.C.F. = 2^□ × 3^□
Fill in each blank
Answer:
7.30 = 2¹ × 3¹ × 5 and 126 = 2¹ × 3² × 7. Their H.C.F. is and their L.C.M. is .
Fill in each blank
Answer:
8.The L.C.M. of 6, 12 and 21 is .
Answer:
9.The L.C.M. of 12, 18 and 20 is .
Answer:
10.60 = 2² × 3¹ × 5 and 168 = 2³ × 3¹ × 7. Their H.C.F. is and their L.C.M. is .
Fill in each blank
Answer:

13.A class of 21 boys and 14 girls splits into groups, each with the same number of boys and the same number of girls. There can be at most groups, each with boys and girls.
Fill in each blank
Working:
Answer:
14.Rectangular tiles 16 cm by 12 cm, all laid the same way, make a square (no tile is cut). What is the fewest tiles needed?
Working:
Answer:
15.Two runners circle a track: one takes 60 seconds a lap, the other 105 seconds. They start together. At least how many seconds later are they both back at the start together?
Working:
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16.A rectangular floor 75 cm by 45 cm is covered with equal square tiles, none cut. The tiles can be at most cm a side, and then tiles are needed.
Fill in each blank
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Answer:
17.Two bus routes leave the terminus together. Route A leaves every 10 minutes and route B every 30 minutes. At least how many minutes later do they next leave together?
Working:
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18.Three alarms ring together, then every 4, 9 and 20 minutes. At least how many minutes later do all three ring together again?
Working:
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Show the answers and working

H.C.F. and L.C.M. worksheet: Answers and working

1. 15
90 = 15 × 6 and 105 = 15 × 7, and 6 and 7 share no factor, so the H.C.F. is 15.
2. 126
14 = 2 × 7 and 18 = 2 × 9, so the L.C.M. is 2 × 7 × 9 = 126.
3. 11
33 = 11 × 3 and 55 = 11 × 5, and 3 and 5 share no factor, so the H.C.F. is 11.
4. 1, 2
For the L.C.M., take each prime's larger index: 2 to the 1, 5 to the 2.
5. 3, 2
For the H.C.F., take each prime's smaller index: 2 to the 3, 3 to the 2.
6. 1, 2
For the H.C.F., take each prime's smaller index: 2 to the 1, 3 to the 2.
7. 6, 630
5 and 7 each belong to one number only: not common, so not in the H.C.F., though the L.C.M. takes them. The H.C.F. takes 2 and 3 to the smaller index: 6. The L.C.M. takes every prime to the larger: 630.
8. 84
Break all three into primes and multiply each prime to its largest index: 84.
9. 180
Break all three into primes and multiply each prime to its largest index: 180.
10. 12, 840
5 and 7 each belong to one number only: not common, so not in the H.C.F., though the L.C.M. takes them. The H.C.F. takes 2 and 3 to the smaller index: 12. The L.C.M. takes every prime to the larger: 840.

11. A. 180
36 = 12 × 3, 60 = 12 × 5, so the L.C.M. is 12 × 3 × 5 = 180.
12. False
40 = 10 × 4 and 50 = 10 × 5: the H.C.F. is 10. A smaller common factor isn't the highest.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 7, 3, 2
The number of groups must divide both 21 and 14; the most is their H.C.F., 7. Each has 21 ÷ 7 = 3 boys and 14 ÷ 7 = 2 girls.
14. 12 tiles
The square's side must be a multiple of both 12 and 16; the least is the L.C.M., 48 cm. That is 48 ÷ 16 tiles a row and 48 ÷ 12 rows: 12 tiles.
15. 420 seconds
The time must be a multiple of both 60 and 105; the least is their L.C.M., 420 seconds.
16. 15, 15
The side must divide both 75 and 45; the largest is their H.C.F., 15 cm. That is (75 ÷ 15) × (45 ÷ 15) = 5 × 3 = 15 tiles.
17. 30 minutes
It must be a multiple of both 10 and 30; the least is their L.C.M., 30 minutes.
18. 180 minutes
It must be a multiple of 4, 9 and 20: the least is their L.C.M., 180.

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