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Sample Question 14:
Step 1: Since we want \(n\) to be as large as possible, we aim to make \(d\) in \(10d+e\) as large as possible. The largest single-digit prime number is 7, so \(d=7\).
Step 2: The next step is to find \(e\) which cannot be 5 because \(10(7)+5=75\) is not a prime number. Therefore, \(e\) should be the next largest single-digit prime number, which is 3. Thus, \(e=3\).
Step 3: Substitute \(d\) and \(e\) into the expression \(d \cdot e \cdot (10d + e)\) to find \(n\). That is \(n = 7 \cdot 3 \cdot 73 = 1533\).
Step 4: Finally, add up the individual digits of \(n\) to get the final answer. The sum of the digits is \(1 + 5 + 3 + 3 = 12\). So, the answer is \(\mathrm{(A) \ } 12\).
Let \(n\) be the largest integer that is the product of exactly 3 distinct prime numbers \(d\), \(e\), and \(10d+e\), where \(d\) and \(e\) are single digits. What is the sum of the digits of \(n\)?
\(\mathrm{(A) \ } 12\qquad \mathrm{(B) \ } 15\qquad \mathrm{(C) \ } 18\qquad \mathrm{(D) \ } 21\qquad \mathrm{(E) \ } 24\)
Answer Keys
Question 14: ASolutions
Question 14Step 1: Since we want \(n\) to be as large as possible, we aim to make \(d\) in \(10d+e\) as large as possible. The largest single-digit prime number is 7, so \(d=7\).
Step 2: The next step is to find \(e\) which cannot be 5 because \(10(7)+5=75\) is not a prime number. Therefore, \(e\) should be the next largest single-digit prime number, which is 3. Thus, \(e=3\).
Step 3: Substitute \(d\) and \(e\) into the expression \(d \cdot e \cdot (10d + e)\) to find \(n\). That is \(n = 7 \cdot 3 \cdot 73 = 1533\).
Step 4: Finally, add up the individual digits of \(n\) to get the final answer. The sum of the digits is \(1 + 5 + 3 + 3 = 12\). So, the answer is \(\mathrm{(A) \ } 12\).