# GCF and LCM

Pro Problems > Math > Number and Quantity > Number Theory > Divisors## GCF and LCM

The greatest common factor of two positive integers is 12, and their least common multiple is 72. If the sum of the numbers is less than 72, what are the numbers?

## Solution

In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. If you would like to view the solutions to these problems, you must have a Virtual Classroom subscription.## Similar Problems

### Find the Missing Number

What number is missing from the following list?

1, 2, 3, 4, 6, 8, 9, 18, 24, 36, 72

### No-Calculator Factoring

Reduce the following number to its prime factorization, without use of a calculator: 1006005. Explain the steps in your process.

### Divisor Count

x has 2n - 9 more divisors than x - 4, and it has 2n - 7 divisors more than x + 5.

What are the possible values of n, if x is a positive integer less than 50?

### Leaping up the Obelisk

An obelisk has no more than 100 stairs. If I go up the stairs two at a time, there will be one step left over at the top. If I go up the stairs three at a time, there will be two left at the top. If I go up the stairs seven at a time, there will be six stairs left over at the top. What is the maximum number of stairs the obelisk could have?

### Two, Three, but not Six

How many integers between 1 and 100 inclusive are multiples of either 2 or 3, but not of 6?

### LCM and GCF

The least common multiple of x and 36 is 180. The greatest common factor of x and 36 is 4. What is the value of x?

### Donut Division

I had enough donuts to share evenly among twelve people without splitting any donuts. Then someone stole sixteen donuts, leaving me with enough to share evenly among seven people. If instead of sharing among seven people, I shared the remaining donuts among four people, how many did each person get, assuming I started with fewer than 100 donuts?

### Multiples of Two and Three

How many integers between 1 and 1000 inclusive are multiples of either 2 or 3?