2018 AMC 8
Complete problem set with solutions and individual problem pages
How many positive three-digit integers have a remainder of when divided by , a remainder of when divided by , and a remainder of when divided by ?
- A.
- B.
- C.
- D.
- E.
Solution 1
Looking at the values, we notice that , and . This means we are looking for a value that is four less than a multiple of , , and . The least common multiple of these numbers is , so the numbers that fulfill this can be written as , where is a positive integer. This value is only a three-digit integer when is or , which gives and respectively. Thus, we have values, so our answer is .
Solution 2
Let us create the equations: , and we know , it gives us , which is the range of the value of z. Because of , then , so must be a multiple of 6. Because of , then , so must also be a multiple of . Hence, the value of must be a common multiple of and , which means multiples of . So, let's say ; then, , so . Thus, the answer is .
