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24 Nov 2021, 07:20
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If \(x = 2^{16}*3^3*5^{18} + 9^{15}\), then how many digits does \(x\) have?
A. \(16\)
B. \(17\)
C. \(18\)
D. \(19\)
E. \(20\)
A. \(16\)
B. \(17\)
C. \(18\)
D. \(19\)
E. \(20\)
24 Nov 2021, 07:20
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Official Solution:
If \(x = 2^{16}*3^3*5^{18} + 9^{15}\), then how many digits does \(x\) have?
A. \(16\)
B. \(17\)
C. \(18\)
D. \(19\)
E. \(20\)
First, let's determine the number of digits in the first term:
\(2^{16}*3^3*5^{18}=\)
\(=(2^{16}*5^{16})*3^3*5^2=\)
\(=10^{16}*675\)
This number will have 19 digits, consisting of 675 followed by 16 zeros.
Next, we need to determine if adding \(9^{15}\) will increase the number of digits. Notice that \(9^{15} < 10^{15}\). If we replace the second term with \(10^{15}\), which is greater than \(9^{15}\), we would have:
\(2^{16}*3^3*5^{18}+10^{15}=\)
\(=10^{16}*675+10^{15}=\)
\(=10^{15}(10*675+1)=\)
\(=10^{15}*6751\).
This number will also have 19 digits, specifically 6751 followed by 15 zeros.
Since adding \(10^{15}\), which is greater than \(9^{15}\), does not increase the number of digits, it implies that adding \(9^{15}\) will not increase the number of digits either. Therefore, the number \(x = 2^{16}*3^3*5^{18} + 9^{15}\) will also have 19 digits.
Answer: D
If \(x = 2^{16}*3^3*5^{18} + 9^{15}\), then how many digits does \(x\) have?
A. \(16\)
B. \(17\)
C. \(18\)
D. \(19\)
E. \(20\)
First, let's determine the number of digits in the first term:
\(2^{16}*3^3*5^{18}=\)
\(=(2^{16}*5^{16})*3^3*5^2=\)
\(=10^{16}*675\)
This number will have 19 digits, consisting of 675 followed by 16 zeros.
Next, we need to determine if adding \(9^{15}\) will increase the number of digits. Notice that \(9^{15} < 10^{15}\). If we replace the second term with \(10^{15}\), which is greater than \(9^{15}\), we would have:
\(2^{16}*3^3*5^{18}+10^{15}=\)
\(=10^{16}*675+10^{15}=\)
\(=10^{15}(10*675+1)=\)
\(=10^{15}*6751\).
This number will also have 19 digits, specifically 6751 followed by 15 zeros.
Since adding \(10^{15}\), which is greater than \(9^{15}\), does not increase the number of digits, it implies that adding \(9^{15}\) will not increase the number of digits either. Therefore, the number \(x = 2^{16}*3^3*5^{18} + 9^{15}\) will also have 19 digits.
Answer: D
04 May 2023, 00:30
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
