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To encourage public transportation, Leo wants to gift some of his friends envelopes with bus tickets and subway tickets in them.
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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 make?
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In order to solve this problem, we need to consider the highest common factor.
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This is the highest number that divides exactly into both of our numbers.
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Leo has 32 bus tickets and 80 subway tickets.
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We need to find the highest common factor of 32 and 80.
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There were lots of ways of finding the factors of a number.
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Firstly, we will look at the method of finding factor pairs.
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Factor pairs are two numbers that multiply together to give the number.
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In this case, we’re looking for pairs of numbers that multiply to give us 32.
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One multiplied by 32 is equal to 32.
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Therefore, one and 32 are factors of 32.
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Two multiplied by 16 also equals 32.
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Therefore, these two numbers are also factors of 32.
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The final factor pair of 32 is four and eight, as four multiplied by eight is equal to 32.
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In ascending order, the factors of 32 are one, two, four, eight, 16, and 32.
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We can now use the same method to find the factors of 80.
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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.
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In ascending order, once again, the factors of 80 are one, two, four, five, eight, 10, 16, 20, 40, and 80.
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The highest number that appears in both of these lines is 16.
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Therefore, the highest common factor of 32 and 80 is equal to 16.
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This means that the greatest number of envelopes that Leo can make is 16.
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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.
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An alternative method to find the factors of 32 and 80 is using prime factor decomposition.
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This involves splitting a number into the product of its prime factors.
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32 can be split into two and 16, as two multiplied by 16 is equal to 32.
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Two is a prime number, so we circle it.
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We now need to split 16.
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16 is equal to two multiplied by eight.
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Once again, two is a prime number.
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Our next step is to split the number eight.
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Eight can be split into two and four, as two multiplied by four is equal to eight.
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Finally, we can split four into two multiplied by two.
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This means that 32 is equal to two multiplied by two multiplied by two multiplied by two multiplied by two.
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This can be simplified to two to the power of five.
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We can split 80 into a product of its prime factors in the same way.
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Two multiplied by 40 is equal to 80.
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Two multiplied by 20 is equal to 40.
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Two multiplied by 10 is equal to 20.
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And finally, two multiplied by five is equal to 10.
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This means that 80 can be written as two to the power of four multiplied by five.
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This proves that the highest common factor of 32 and 80 is two to the power of four.
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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.
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Therefore, once again, the highest number of envelopes that Leo can make is 16.