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`Highest‐Common‐Factors and Lowest‐Common‐Multiples 1. For each pair of numbers, draw a factor sharing diagram and use it to determine the highest common factor of the two numbers: (a) 24 and 120 (b) 102 and 255 (c) 63 and 275 (d) 1024 and 64 2. Here is the factor sharing diagram for two numbers, X and Y: X 3 5 11 2 7 2 Y (a) What is the HCF of X and Y? (b) What are the numbers X and Y? (c) The prime factorisation of X is 22×3×11. Write down the prime factorisation of Y. (d) Write down the prime factorisations of the following: (i) 3X (ii) 5X (iii) 7X (iv) 4X (v) 6X (vi) 15X (e) Which of the following are multiples of X: for those that are multiples of X, state what multiple (Hint ‐ e.g. 25×3×11 = 22X, so 4X) (i) 26×3×11 (ii) 22×3×13 (v) 22×3×7 (vi) 22×3×5×7×11 (iii) 22×33×11 (iv) 2×32×11 (vii) 24×32×5×7×11 (viii) 23×32×112 (f) Write down the prime factorisations of the following: (i) 3Y (ii) 5Y (iii) 11Y (iv) 4Y (v) 6Y (vi) 15Y (g) Which of the following are multiples of Y (i) 26×3×5×72 (ii) 2×3×5×7 (iii) 22×33×11×7 (iv) 23×3×53×7 (v) 22×3×7×11 (vi) 22×3×5×7×11 (vii) 24×32×5×7×11 (viii) 24×32×5×74 (h) Look at your two lists for questions (e) and (g). There are two numbers which are both a multiple of X and a multiple of Y. Write down the numbers in factorised form and then use your calculator to work out what they are. (i) What is the Lowest Common Multiple of X and Y? 3. Here is the factor sharing diagram for two numbers P and Q. P 5 3 Q 3 2 (a) What is the number P? (b) What is the HCF of P and Q? (c) Q is in fact the number 84. What factors are missing from Q? (d) Copy and complete this table to show which numbers are multiples of P and Q: Multiple of P? Multiple of Q? 24×3×7 22×33×5 25×33×5×72 3×5×7 22×32×5×7 23×32×52×7 (e) What is the lowest common multiple of P and Q? 4. By looking at our factor sharing diagrams, find the LCM of each of the pairs of numbers in Question 1. 5*. (a) Find the prime factorisations of 60, 280 and 300. (b) Copy and complete the Factor sharing diagram for 60, 280 and 300 shown below: 60 280 (ii) 300 and 280 (iii) 60 and 280 (ii) 300 and 280 (iii) 60 and 280 300 (c) What is the HCF of: (i) 300 and 60 (iv) 300 and 60 and 280 (d) What is the LCM of: (i) 300 and 60 (iv) 300 and 60 and 280 `