Question7 8 (a) Express 918 as the product of its prime factors. (b) Given that $1275=3 \times 5^{2} \times 17$, find the highest common factor (HCF) of 918 and 1275 . $918: 2$ (3)(17) Answer $918=2 \times 3^{3} \times 17$ $\qquad$ $[1]$ $1275,35,0$
Answer $\qquad$ 17 $\qquad$ [1] (c) Thumbtacks are sold in boxes of 918. Paper clips are sold in boxes of 1275. Tim bought the same number of paper clips and thumbtacks. Find the least number of boxes of paper clips he could have bought. Answer [2]

Studdy Solution
Determine the least number of boxes of paper clips Tim could have bought to match the number of thumbtacks.
To find the least number of boxes, we need to find the least common multiple (LCM) of 918 and 1275.
The LCM is obtained by taking the highest power of all prime factors present in both numbers: - $2^1$ - $3^3$ - $5^2$ - $17^1$
Thus: $$LCM = 2^1 \times 3^3 \times 5^2 \times 17$$
Calculating this: $$LCM = 2 \times 27 \times 25 \times 17$$ $$LCM = 2 \times 675 \times 17$$ $$LCM = 1350 \times 17$$ $$LCM = 22950$$
Since 22950 is the total number of thumbtacks and also the total number of paper clips, we divide by the number of items per box to find the number of boxes.
For thumbtacks: $$\text{Number of boxes of thumbtacks} = \frac{22950}{918} = 25$$
For paper clips: $$\text{Number of boxes of paper clips} = \frac{22950}{1275} = 18$$
Thus, the least number of boxes of paper clips that Tim could have bought is: $$18$$