To encourage public transportation, Leo wants to gift some of his friends envelopes
with bus tickets and subway tickets in them. If he has 32 bus tickets and 80 subway tickets to split equally among the envelopes
and wants no tickets left over, what is the greatest number of envelopes Leo can
In order to solve this problem, we need to consider the highest common factor. This is the highest number that divides exactly into both of our numbers. Leo has 32 bus tickets and 80 subway tickets. We need to find the highest common factor of 32 and 80.
There were lots of ways of finding the factors of a number. Firstly, we will look at the method of finding factor pairs. Factor pairs are two numbers that multiply together to give the number. In this case, we’re looking for pairs of numbers that multiply to give us 32. One multiplied by 32 is equal to 32. Therefore, one and 32 are factors of 32. Two multiplied by 16 also equals 32. Therefore, these two numbers are also factors of 32. The final factor pair of 32 is four and eight, as four multiplied by eight is equal
to 32. In ascending order, the factors of 32 are one, two, four, eight, 16, and 32.
We can now use the same method to find the factors of 80. There are five factor pairs of 80: one and 80, two and 40, four and 20, five and 16,
and eight and 10, as all five of these pairs multiply together to give us 80. In ascending order, once again, the factors of 80 are one, two, four, five, eight,
10, 16, 20, 40, and 80.
The highest number that appears in both of these lines is 16. Therefore, the highest common factor of 32 and 80 is equal to 16. This means that the greatest number of envelopes that Leo can make is 16. Each envelope would have two bus tickets and five subway tickets, as 16 multiplied by
two is equal to 32 and 16 multiplied by five is equal to 80.
An alternative method to find the factors of 32 and 80 is using prime factor
decomposition. This involves splitting a number into the product of its prime factors. 32 can be split into two and 16, as two multiplied by 16 is equal to 32. Two is a prime number, so we circle it.
We now need to split 16. 16 is equal to two multiplied by eight. Once again, two is a prime number. Our next step is to split the number eight. Eight can be split into two and four, as two multiplied by four is equal to
eight. Finally, we can split four into two multiplied by two. This means that 32 is equal to two multiplied by two multiplied by two multiplied by
two multiplied by two. This can be simplified to two to the power of five.
We can split 80 into a product of its prime factors in the same way. Two multiplied by 40 is equal to 80. Two multiplied by 20 is equal to 40. Two multiplied by 10 is equal to 20. And finally, two multiplied by five is equal to 10. This means that 80 can be written as two to the power of four multiplied by five.
This proves that the highest common factor of 32 and 80 is two to the power of
four. And as two to the power of four is equal to 16, we can say that the highest common
factor of 32 and 80 is 16. Therefore, once again, the highest number of envelopes that Leo can make is 16.