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