Test Answering Strategy
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?
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
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, $ 4 $ and $ 8 $ satisfy Case 1 and $ 3,6, $ and $ 9 $ 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 $ 8 \cdot 5 = 40 $ questions correctly. However, remember that we must also consider the leftover numbers 97, 98, 99, 100 and out of these, $ 99 $ and $ 100 $ satisfy one of the cases. So our final number of correct answers is $ 40 + 2 = \boxed{42} $.
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