Biscuit packets contain 12, 15 and 21 biscuits. Find the minimum number of packets of each brand to buy an equal number of biscuits of each brand.
\(35\) packets of 12, \(28\) packets of 15, and \(20\) packets of 21.
We must get the same total biscuits from each brand. Use the Least Common Multiple (LCM) of 12, 15, and 21.
Write prime factors.
\(12 = 2 \times 2 \times 3 = 2^2 \times 3\)
\(15 = 3 \times 5\)
\(21 = 3 \times 7\)
Take each prime with the highest power.
\(\text{LCM} = 2^2 \times 3 \times 5 \times 7\)
\(= 4 \times 3 \times 5 \times 7\)
\(= 420\)
So we will match at \(420\) biscuits for each brand.
Find packets for each brand.
12-biscuit brand: \(420 \div 12 = 35\) packets
15-biscuit brand: \(420 \div 15 = 28\) packets
21-biscuit brand: \(420 \div 21 = 20\) packets
Minimum required: \(35\), \(28\), and \(20\) packets respectively.