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A florist has 91 red roses and 78 pink roses.
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He wants to make identical bunches using all his flowers.
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What is the largest number of bunches he can make?
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As the florist wants to make identical bunches, we need to have the same number of red roses and the same number of pink roses in each of the bunches.
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In order to do this, we need to firstly find the highest common factor of 91 and 78.
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One way of doing this is using prime factorization.
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Prime numbers are numbers with exactly two factors, the numbers one and itself.
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The prime numbers less than 20 are two, three, five, seven, 11, 13, 17, and 19.
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Two is not a factor of 91, as 91 is not an even number.
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Three is also not a factor of 91.
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Any integer is divisible by three if the sum of its digits is also divisible by three.
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The sum of the digits in 91 is 10, and as this is not divisible by three, then 91 will not be divisible by three.
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The only numbers that are divisible by five have a five or zero in the units column.
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As 91 does not, it will not be divisible by five.
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91 is divisible by seven, as seven multiplied by 13 is equal to 91.
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Seven multiplied by 10 is equal to 70.
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Seven multiplied by three is equal to 21.
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Adding 70 and 21 gives us 91.
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This means that 91 written as a product of its prime factors is seven multiplied by 13.
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We can repeat this process for 78.
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As 78 is even, it must be divisible by two.
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Two multiplied by 39 is equal to 78, as a half of 78 is 39.
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39 is not a prime number.
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It is divisible by three, as three multiplied by 13 is equal to 39.
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78 written as a product of its prime factors is two multiplied by three multiplied by 13.
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The only prime number that is common is 13.
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Therefore, the highest common factor of 91 and 78 is 13.
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This means that the florist could make 13 bunches.
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Each of the bunches would have seven red roses, as seven multiplied by 13 is 91, and six pink roses, as six multiplied by 13 is equal to 78.
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The largest number of bunches he can make is 13.