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Sample Question 13:
Step 1: Identify the form of three-digit numbers we're looking for \(xyx\). With this, \(x\) cannot equal 0 (which would result in a two-digit integer), and \(x\) cannot equal 5 (which would create a number divisible by 5).
Step 2: Implementing the second condition that the sum of \(2x+y<20\), if \(x\) equals 1, 2, 3, or 4, \(y\) can be any digit from 0 to 9 because \(2x<10\). This gives us 10 possibilities for each, totalling 40 numbers.
Step 3: If \(x=6\), then we find that \(12+y<20\), implying that \(y<8\). This gives us 8 more numbers.
Step 4: Continuing with \(x=7\), this results in \(14+y<20\), and therefore \(y<6\). This further adds 6 numbers.
Step 5: When \(x=8\), we obtain \(16+y<20\), so \(y<4\), which equates to 4 extra numbers.
Step 6: Lastly, for \(x=9\), the equation translates to \(18+y<20\), giving us \(y<2\), and 2 more numbers.
Step 7: To find the total number of three-digit numbers that meet all conditions, sum all the obtained numbers: \(40 + 8 + 6 + 4 + 2 = 60\) numbers. Hence, the answer is \(\textbf{(B) }60\).
How many three-digit numbers are not divisible by \(5\), have digits that sum to less than \(20\), and have the first digit equal to the third digit?
\(\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 66 \qquad\textbf{(D)}\ 68 \qquad\textbf{(E)}\ 70\)
Answer Keys
Question 13: BSolutions
Question 13Step 1: Identify the form of three-digit numbers we're looking for \(xyx\). With this, \(x\) cannot equal 0 (which would result in a two-digit integer), and \(x\) cannot equal 5 (which would create a number divisible by 5).
Step 2: Implementing the second condition that the sum of \(2x+y<20\), if \(x\) equals 1, 2, 3, or 4, \(y\) can be any digit from 0 to 9 because \(2x<10\). This gives us 10 possibilities for each, totalling 40 numbers.
Step 3: If \(x=6\), then we find that \(12+y<20\), implying that \(y<8\). This gives us 8 more numbers.
Step 4: Continuing with \(x=7\), this results in \(14+y<20\), and therefore \(y<6\). This further adds 6 numbers.
Step 5: When \(x=8\), we obtain \(16+y<20\), so \(y<4\), which equates to 4 extra numbers.
Step 6: Lastly, for \(x=9\), the equation translates to \(18+y<20\), giving us \(y<2\), and 2 more numbers.
Step 7: To find the total number of three-digit numbers that meet all conditions, sum all the obtained numbers: \(40 + 8 + 6 + 4 + 2 = 60\) numbers. Hence, the answer is \(\textbf{(B) }60\).