A mathematics teacher wishes to use groups of 5 and/or 7 students for a class assignment. When she tries to make assignments she notices that it cannot be done with the number of students she has. However, if the number of students were any larger, the assignments could be made. How many students has the teacher?
I can be done with a minimum of 35 students, so she has less than 35 students
To solve this problem, we need to find the smallest number that is not divisible by either 5 or 7.
Let's start by finding the smallest number that is divisible by 5 and 7. This is called the least common multiple (LCM).
To find the LCM of two numbers, we can use the formula:
LCM(a, b) = (a * b) / GCD(a, b)
Here, GCD refers to the greatest common divisor, which is the largest number that divides both a and b evenly.
The GCD of 5 and 7 is 1 because they are both prime numbers. So, the LCM(5, 7) is (5 * 7) / 1 = 35.
Now, let's find the smallest number that is not divisible by 35. Starting from 1, we will go through the numbers one by one until we find the smallest one that is not divisible by 35.
1 is divisible by 35, so we move on to the next number.
2 is also divisible by 35, so we continue.
3 is not divisible by 35, so we have found the answer. The teacher must have 3 students.
If there were any more students, the assignments could be made.