Xavier is a dental sales representative who wants to distribute 30 brochures and 87 pamphlets to local dental offices. He wants to deliver the same combination of brochures and pamphlets to each office, without having any materials left over. What is the greatest number of dental offices Xavier can distribute materials to?
Question
Xavier is a dental sales representative who wants to distribute 30 brochures and 87 pamphlets to local dental offices. He wants to deliver the same combination of brochures and pamphlets to each office, without having any materials left over. What is the greatest number of dental offices Xavier can distribute materials to?
Solution
To find the greatest number of dental offices Xavier can distribute materials to, we need to find the greatest common divisor (GCD) of the number of brochures and pamphlets. The GCD is the largest number that can divide both numbers without leaving a remainder.
Here are the steps to find the GCD of 30 (the number of brochures) and 87 (the number of pamphlets):
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List the prime factors of each number.
- The prime factors of 30 are 2, 3, and 5.
- The prime factors of 87 are 3 and 29.
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Identify the common prime factors.
- The only common prime factor between 30 and 87 is 3.
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The GCD is the product of the common prime factors.
- Since 3 is the only common prime factor, the GCD is 3.
Therefore, the greatest number of dental offices Xavier can distribute materials to is 3.
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