Practice/Help on HCF & LCM Problem Sums

Problem sums on HCF and LCM can be really tricky as they are not easy to identify. Thus for this post, the main focus is not on going through how to find HCF and LCM (please refer to your notes on those), but more importantly to go through how to determine when to find the HCF and when to find the LCM of the numbers involved in the problem sums.

There will be some problem sums for you to try out at the end of the post, but first let’s take a look at a typical problem involving the HCF.

HCF – A Typical Problem

3 strings of different lengths, 240 cm, 318 cm and 426 cm are to be cut into equal lengths. What is the greatest possible length of each piece?

If you notice, finding the HCF is crucial here because you are trying to find what the 3 numbers have in common, i.e. a common factor. All 3 numbers must be able to be divided by the same number in order for all 3 strings to be cut into equal lengths. HCF is needed here because you are asked to find the greatest possible length.

Therefore,

LCM – A Typical Problem

Two lighthouses flash their lights every 20s and 30s respectively. Given that they flashed together at 7pm, when will they next flash together?

One method to finding the next time the lighthouses flash together is:

20, 40, 60

30, 60, 90

60 is a multiple common to 20 and 30, and thus the lighthouses will flash together in 60s’ time, i.e. at 7:01pm.

This is the same as finding the lowest common multiple, or LCM:

There are other different types of problems involving LCM, but just remember that such questions involve you trying to find a multiple that is common to the numbers involved.

Try out the problem sums below and see if you get them right! The starred ones require a little more thinking

1. As a humanitarian effort, food ration is distributed to each refugee in a refugee camp. If a day’s ration is 284 packets of biscuits, 426 packets of instant noodles and 710 bottles of water, how many refugees are there in the camp? [142 refugees]

2. 294 blue balls, 252 pink balls and 210 yellow balls are distributed equally among some students with none left over. What is the biggest possible number of students? [42 students]

3. A group of girls bought 72 rainbow hairbands, 144 brown and black hairbands, and 216 bright-coloured hairbands. What is the largest possible number of girls in the group? [72 girls]

4. A man has a garden measuring 84 m by 56 m. He wants to divide them equally into the minimum number of square plots. What is the length of each square plot? [28 m]

5. Leonard wants to cut identical square as big as he can from a piece of paper 168 mm by 196 mm. What is the length of each square? [28 cm]

6.* 32 girls and 52 boys were on an overseas learning trip, and they were divided into as many groups as possible where the number of groups of girls and the number of groups of boys are the same. How many girls and how many boys are there in each group? [8 girls, 13 boys]

7. A small bus interchange has 2 feeder services that start simultaneously at 9am. Bus number 801 leaves the interchange at 15-min intervals, while bus number 802 leaves at 20-min intervals. On a particular day, how many times did both services leave together from 9 am to 12 noon inclusive? [4 times]

8. Candice, Gerald and Johnny were jumping up a flight of stairs. Candice did 2 steps at a time, Gerald 3 steps at time while Johnny 4 steps at a time. If they started on the bottom step at the same, on which step will all 3 land together the first time? [12th step]

9. Heidi helps out at her mum’s stall every 9 days while her sister every 3 days. When will they be together if they last helped out on June 16 2008? [June 25 2008]

10. A group of students can be further separated into groups of 5, 13 and 17. What is the smallest possible total number of students? [1105 students]

11. Jesslyn goes to the market every 64 days. Christine goes to the same market every 72 days. They met each other one day. How many days later will they meet each other again? [576 days]

12.*Mrs Goh and 3 of her friends went to a supermarket and found that a package of 6 dishcloths cost $10. If they were to share the purchase such that each has the same number of dishcloths, what is the minimum amount each has to pay? [$5]

1. Write the prime factorisation of the greatest 3-digit number.

2. Write the prime factorisation of the following numbers in exponential form:(i) 13860 (ii) 27830 (iii) 21952.

3. Find the smallest number which must be added to 9373 so that it becomes divisible by 4.

4. Find the smallest number which must be added to 605329 so that it becomes divisible by 9.

5. Replace the letter x in the number 8x516 by the smallest digit so that the number becomes divisible by 6.

6. Find the H.C.F. of:(i) 24, 60, 112        (ii) 70, 84, 336, 1260.

7. Find the least number which on adding 7 is exactly divisible by each of 15, 35 and 48.

8. Find the greatest number of four digits which is exactly divisible by each of 12, 18, 40 and 45.

9. Find the least number of five digits which is exactly divisible by each of 32, 36, 60, 90 and 144.

10. Find the H.C.F. of:(i) 72, 126, 168      (ii) 96, 528, 2160, 3520.

11. Find the greatest number that will divide 400, 435 and 541 leaving 9, 10 and 14 as remainders respectively.

12. Which of the following pairs of numbers are co-prime?(i) 15, 98      (ii) 198, 429     (iii) 847, 2160

13. If the product of two numbers is 84942 and their H.C.F. is 33, find their L.C.M.

14. The product of H.C.F. and L.C.M. of two numbers is 9072. If one of the numbers is 72, find the other number.

15. The H.C.F. and L.C.M. of two numbers are 12 and 5040 respectively If one of the numbers is 144, find the other number.

Answers

1. 999 = 3³×372. (i) 2²×3²×5×7×11    (ii) 2×5×11²×23   (iii) 26×7³3. 3           4. 2            5. 1             6. (i) 4   (ii) 147. 1673     8. 9720     9. 10080     10. (i) 6    (ii) 16       11. 1712. (i) Co-prime           (ii) not co-prime           (iii) co-prime13. 2574   14. 126       15. 420