Mistake in your working.

450 = 2 x 3² x 5², not 2 x 3 x 5²

GCF or HCF = 2 x 3² = 18

The question is poorly phrased.

If we follow a) and find the maximum number of packets such that each packet has an equal quantity of each colour,

then one just needs to put only 1 of each colour into every packet.


The number green buttons is the fewest of the three kinds.

This means the most number of packets we can have is 126 since :

126 green buttons ÷ 1 green button per packet = 126 packets

This means that each packet has 1 blue , 1 green and 1 red button only

What the question is actually trying to ask is :

To find the maximum number of packets,
such that the number of buttons of the same colour is the same for every packet.

In this case from a) we know that the HCF is 18

So there can be a maximum of 18 packets.

Number of blue buttons per packet

= Number of blue buttons in total ÷ number of packets

= 396 ÷ 18
= 22

Alternatively,

396 = 2² x 3² x 11

= 2 x 3² x 2 x 11

= 18 x 22

So there must be 18 packets where each packet contains 22 blue buttons