Sports day organisers have 144 medals to pack into identical boxes, with a prime number of medals in each box and none left over. Express 144 as a product o...
Question 1 Report
Sports day organisers have 144 medals to pack into identical boxes, with a prime number of medals in each box and none left over.
- Express 144 as a product of prime numbers, using index form for your answer. (3)
- Write down the greatest prime number that could be the number of medals in each box. (1)
Answer Details
This question tests writing a number as a product of prime factors in index form, and reasoning about which of those prime factors is the largest.
(a) Dividing 144 repeatedly by prime numbers: \(144 = 2 \times 72 = 2 \times 2 \times 36 = 2 \times 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 2 \times 9 = 2^4 \times 3 \times 3\). Collecting the prime factors:
\[144 = 2^4 \times 3^2\][3] (1 for a correct factor tree or repeated division, 1 for identifying all the prime factors, 1 for correct index form)
(b) A prime number of medals per box must be a prime factor of 144. The only prime factors of 144 are 2 and 3, so the greatest possible prime number of medals per box is
\[3\][1]
Once a number is written in index form, its prime factors can be read off directly as the bases of the powers; here only 2 and 3 appear, so no larger prime such as 5 or 7 could ever divide 144 exactly.
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