Browse · MATH
Printjmc
number theory senior
Problem
On a true-false test of 100 items, every question that is a multiple of 4 is true, and all others are false. If a student marks every item that is a multiple of 3 false and all others true, how many of the 100 items will be correctly answered?
Solution
The student will answer a question correctly if
Case 1: both the student and the answer key say it is true. This happens when the answer is NOT a multiple of 3 but IS a multiple of 4.
Case 2. both the student and the answer key say it is false. This happens when the answer IS a multiple of 3 but is NOT a multiple of 4.
Since the LCM of 3 and 4 is 12, the divisibility of numbers (in our case, correctness of answers) will repeat in cycles of 12. In the first 12 integers, and satisfy Case 1 and and satisfy Case 2, so for every group of 12, the student will get 5 right answers. Since there are 8 full groups of 12 in 100, the student will answer at least questions correctly. However, remember that we must also consider the leftover numbers 97, 98, 99, 100 and out of these, and satisfy one of the cases. So our final number of correct answers is .
Case 1: both the student and the answer key say it is true. This happens when the answer is NOT a multiple of 3 but IS a multiple of 4.
Case 2. both the student and the answer key say it is false. This happens when the answer IS a multiple of 3 but is NOT a multiple of 4.
Since the LCM of 3 and 4 is 12, the divisibility of numbers (in our case, correctness of answers) will repeat in cycles of 12. In the first 12 integers, and satisfy Case 1 and and satisfy Case 2, so for every group of 12, the student will get 5 right answers. Since there are 8 full groups of 12 in 100, the student will answer at least questions correctly. However, remember that we must also consider the leftover numbers 97, 98, 99, 100 and out of these, and satisfy one of the cases. So our final number of correct answers is .
Final answer
42