Greatest Common Factor Problems – Civil Service Exam

Greatest Common Factor Problems - Civil Service Exam

Problem 1:

A caterer has 180 cupcakes and 240 mini pies. She wants to create dessert platters that have the same number of cupcakes and mini pies without any leftovers. What is the maximum number of platters she can create?

Answer: 60 platters

Solution: To find the maximum number of platters the caterer can create, calculate the GCD of 180 and 240.

Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
Common factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 The GCD is 60. Therefore, she can create 60 dessert platters.

Problem 2:

A tailor has 294 buttons and 462 zippers. She wants to create kits that contain an equal number of buttons and zippers without any leftovers. What is the maximum number of kits she can make?

Answer: 42 kits

Solution: To determine the maximum number of kits, calculate the GCD of 294 and 462.

Factors of 294: 1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, 294
Factors of 462: 1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462
Common factors: 1, 2, 3, 6, 7, 14, 21, 42 The GCD is 42. Therefore, she can make 42 kits.

Problem 3:

A school has 132 textbooks and 308 notebooks to be distributed in student packs. Each pack must contain the same number of textbooks and notebooks without any leftovers. What is the maximum number of student packs they can prepare?

Answer: 44 packs

Solution: To find the maximum number of student packs, calculate the GCD of 132 and 308.

Factors of 132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132
Factors of 308: 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308
Common factors: 1, 2, 4, 11, 22, 44 The GCD is 44. Therefore, they can prepare 44 student packs.

When do we use GCF and LCM

LCM Problems

The Least Common Multiple (LCM) of two or more integers is the smallest integer that is divisible by all the numbers without any remainder. LCM problems typically involve situations where you need to find a common time or quantity that involves multiples of given numbers synchronizing or matching up. Here’s why the problems I provided are LCM problems:

Synchronization: When you are looking for a point in time or a measure where events align or repeat together, you are seeking the LCM. For instance, when two people starting exercises at different intervals want to find out when they will next exercise together, they are looking for the LCM of their exercise intervals. This is because the LCM represents the smallest interval at which both original intervals will align.
Repetition or Alignment: Similarly, in scenarios involving traffic lights changing at different intervals or episodes of TV shows of different lengths finishing at the same time, you seek the LCM to find the earliest time both events will coincide.

GCD Problems

The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF), is the largest number that divides two or more integers without leaving a remainder. Here are scenarios typically associated with GCD:

Dividing into groups or parts: If you are dividing a set of items into smaller groups and need to know the largest group size that allows all items to be evenly distributed without leftovers, you use the GCD. For example, if you have a certain number of pencils and markers and want to package them in such a way that each package has the same number of pencils and markers without any left over, you look for the GCD of the number of pencils and markers.
Reduction or Simplification: When reducing fractions to their simplest form, you divide the numerator and the denominator by their GCD.

In summary, use LCM when you need to synchronize or align intervals or sequences, and use GCD when dividing things into uniform groups without leftovers or simplifying ratios. This distinction helps apply the right mathematical concept to practical problems effectively.