Sample Question 12:

How many three-digit positive integers \(N\) satisfy the following properties?

• The number \(N\) is divisible by \(7\).

• The number formed by reversing the digits of \(N\) is divisible by \(5\).

\(\textbf{(A) } 13 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17\)

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Answer Keys

Question 12: B

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Solutions

Question 12
Step 1: Understand and Apply the Properties
We know multiples of \(5\) will always end in \(0\) or \(5\). However, since the number can't start with 0 (otherwise it would be a two-digit number), we're only left with three-digit numbers starting with \(5\). For these numbers to be divisible by 7, they should fall in the range from \(7 \times 72\) to \(7 \times 85\) inclusive, because those are the possible multiples of 7 with a leading digit of 5.

Step 2: Calculate the Possible Integers
We deduct the smallest possible multiple of 7 in our range, \(7 \times 72\), from the largest, \(7 \times 85\), to find the number of multiples of 7 (hence the number of possible three-digit integers \(N\)). We add 1 because both boundaries are inclusive. Thus, \(85 - 72 + 1 = 14\).

Step 3: Answer the Problem
So, the number of three-digit positive integers \(N\) that satisfy the properties is 14. Therefore, the answer is \(\textbf{(B) } 14\).