A math 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 in her class. However, if the number of students were any larger, the assignments could be made. How many students does the teacher have?
Let's look at the multiples of these numbers.
5, 10, 15, 20, 25, 30, 35, 40
7, 14, 21, 28, 35, 42, 49
The LCM is 35
Do you suppose that the teacher has 34 in the class?
To find the number of students the teacher has, we need to find the least common multiple (LCM) of 5 and 7. The LCM is the smallest number that is divisible by both 5 and 7.
To find the LCM, we can use the prime factorization method.
The prime factors of 5 are: 5
The prime factors of 7 are: 7
The LCM is calculated by taking all the unique prime factors and raising each to the highest power:
LCM = (5^1) * (7^1) = 5 * 7 = 35
Therefore, the teacher has 35 students.
To solve this problem, we need to find the smallest number that cannot be divided equally by either 5 or 7. This can be done by finding the least common multiple (LCM) of 5 and 7 and subtracting 1 from it.
The LCM of 5 and 7 is 35 since both numbers are prime and do not share any common factors other than 1. Subtracting 1 from the LCM, we get 34.
Therefore, the teacher must have 34 students in her class. If there were any more students, the assignments could be made using groups of 5 or 7.