Ws2 Factors Multiples
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- Words: 534
- Pages: 2
Date:
______/______/ 2009
Marks
Worksheet 2: Factors and Multiples (HCF and LCM) 1
Basic Question Given that 198 = 2 × 32 × 11, and 90 = 2 × 32 × 5 , find (a) the LCM of 198 and 90. (b) the smallest integer, k, such that 198k is a perfect square. Solution: (a)
2
(b) 198k = 2 × 32 × 11× k ∴ k = 2 × 11 = 22 [Xin Min Sec/2007] Ans: (a) 990 (b) 22
LCM = 2 × 32 × 5 ×11 = 990
Intermediate Question Find the highest common factor (HCF) and lowest common multiple (LCM) of 1 260 and 2 3 × 3 × 7 2 × 112 . Express your answers in index notation. Solution:
1260 = 2 2 × 32 × 5 × 7 and
23 × 3
× 7 2 × 112
HCF = 22 × 3 × 7 LCM = 23 × 32 × 5 × 7 2 × 112 [Cedar Girls’ Sec School/2008] Ans: HCF = 2 × 3 × 7 , LCM = 23 × 32 × 5 × 7 2 × 112 2
3
Intermediate Question Three ribbons of lengths 160 cm, 192 cm and 240 cm respectively are to be cut into a number of equal lengths without any leftover. Find (a) the greatest possible length of each piece and (b) the total number of pieces cut from the three ribbons. Solution: Greatest possible length = 16 cm Total number of pieces = 10 + 12 + 15 = 37
2 2 2 2
160 80 40 20 10
192 96 48 24 12
240 120 60 30 15 Ans: (a) 16 cm (b) 37
© Victoria School Math Department
4
Advance Question Given that a = 22 × 33 × 54 and b = 32 × 4 2 × 52 , find (i) the HCF and (ii) the LCM of a and b. Solution: a = 22 × 33 × 54 b = 24 × 32 × 52 since 42 = (2 × 2) 2 = 24 ∴ HCF = 22 × 32 × 52 = 900 ∴ LCM = 24 × 33 × 54 = 270000 Ans: (a) 900 (b) 270 000
5
Advance Question At East Coast Park, BBQ pits are found at intervals of 6 metres along the beach. At every interval of 8 metres and 15 metres, there is a bench and lamp post respectively. At the beginning of the beach, all the three objects are in line and the BBQ pit number is 1. (a) At what distance will all the objects appear in line for the second time? What is the BBQ pit number when this happens? (b) If the stretch of beach is half a kilometers long, how many times will all the objects appear in line? Solution: (a)
LCM of 6, 8 and 15 = 120 They will be in line at every 120m. 120 It will be at Pit number = + 1 = 21 6
(b)
At 500 m, the objects will appear for 500 = +1 120 ≈ 4 +1 = 5 times
Ans: (a) 120 m, 21 (b) 5
© Victoria School Math Department
______/______/ 2009
Marks
Worksheet 2: Factors and Multiples (HCF and LCM) 1
Basic Question Given that 198 = 2 × 32 × 11, and 90 = 2 × 32 × 5 , find (a) the LCM of 198 and 90. (b) the smallest integer, k, such that 198k is a perfect square. Solution: (a)
2
(b) 198k = 2 × 32 × 11× k ∴ k = 2 × 11 = 22 [Xin Min Sec/2007] Ans: (a) 990 (b) 22
LCM = 2 × 32 × 5 ×11 = 990
Intermediate Question Find the highest common factor (HCF) and lowest common multiple (LCM) of 1 260 and 2 3 × 3 × 7 2 × 112 . Express your answers in index notation. Solution:
1260 = 2 2 × 32 × 5 × 7 and
23 × 3
× 7 2 × 112
HCF = 22 × 3 × 7 LCM = 23 × 32 × 5 × 7 2 × 112 [Cedar Girls’ Sec School/2008] Ans: HCF = 2 × 3 × 7 , LCM = 23 × 32 × 5 × 7 2 × 112 2
3
Intermediate Question Three ribbons of lengths 160 cm, 192 cm and 240 cm respectively are to be cut into a number of equal lengths without any leftover. Find (a) the greatest possible length of each piece and (b) the total number of pieces cut from the three ribbons. Solution: Greatest possible length = 16 cm Total number of pieces = 10 + 12 + 15 = 37
2 2 2 2
160 80 40 20 10
192 96 48 24 12
240 120 60 30 15 Ans: (a) 16 cm (b) 37
© Victoria School Math Department
4
Advance Question Given that a = 22 × 33 × 54 and b = 32 × 4 2 × 52 , find (i) the HCF and (ii) the LCM of a and b. Solution: a = 22 × 33 × 54 b = 24 × 32 × 52 since 42 = (2 × 2) 2 = 24 ∴ HCF = 22 × 32 × 52 = 900 ∴ LCM = 24 × 33 × 54 = 270000 Ans: (a) 900 (b) 270 000
5
Advance Question At East Coast Park, BBQ pits are found at intervals of 6 metres along the beach. At every interval of 8 metres and 15 metres, there is a bench and lamp post respectively. At the beginning of the beach, all the three objects are in line and the BBQ pit number is 1. (a) At what distance will all the objects appear in line for the second time? What is the BBQ pit number when this happens? (b) If the stretch of beach is half a kilometers long, how many times will all the objects appear in line? Solution: (a)
LCM of 6, 8 and 15 = 120 They will be in line at every 120m. 120 It will be at Pit number = + 1 = 21 6
(b)
At 500 m, the objects will appear for 500 = +1 120 ≈ 4 +1 = 5 times
Ans: (a) 120 m, 21 (b) 5
© Victoria School Math Department
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