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03 Dec 2019, 00:49
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What is the units digit of \(66^6 - 5^{55}\)?
A. \(0\)
B. \(1\)
C. \(5\)
D. \(6\)
E. \(9\)
A. \(0\)
B. \(1\)
C. \(5\)
D. \(6\)
E. \(9\)
03 Dec 2019, 00:49
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Official Solution:
What is the units digit of \(66^6 - 5^{55}\)?
A. \(0\)
B. \(1\)
C. \(5\)
D. \(6\)
E. \(9\)
A number with the units digit of 6, when raised to any positive integer power, will have 6 as its units digit. Thus, the units digit of \(66^6\) is 6.
A number with the units digit of 5, when raised to any positive integer power, will have 5 as its units digit. Thus, the units digit of \(5^{55}\) is 5.
At this point, it's crucial not to presume that the units digit of \(66^6 - 5^{55}\) will be \(6 - 5 = 1\), because it's important to recognize that \(5^{55}\) is a MUCH larger number than \(66^6\). Why is this so? \(5^{55}=5^{5*11}=(5^5)^{11}\). Now, \(5^5>66\) and \(11>6\), so it follows that \((5^5)^{11}>66^6\).
Therefore, the units digit of \(66^6 - 5^{55}\) will be:
\(\{ smaller \ number \ with \ the \ units \ digit \ of \ 6 \} - \{ larger \ number \ with \ the \ units \ digit \ of \ 5\} = 9\).
For instance, \(16-25=-9\).
Answer: E
What is the units digit of \(66^6 - 5^{55}\)?
A. \(0\)
B. \(1\)
C. \(5\)
D. \(6\)
E. \(9\)
A number with the units digit of 6, when raised to any positive integer power, will have 6 as its units digit. Thus, the units digit of \(66^6\) is 6.
A number with the units digit of 5, when raised to any positive integer power, will have 5 as its units digit. Thus, the units digit of \(5^{55}\) is 5.
At this point, it's crucial not to presume that the units digit of \(66^6 - 5^{55}\) will be \(6 - 5 = 1\), because it's important to recognize that \(5^{55}\) is a MUCH larger number than \(66^6\). Why is this so? \(5^{55}=5^{5*11}=(5^5)^{11}\). Now, \(5^5>66\) and \(11>6\), so it follows that \((5^5)^{11}>66^6\).
Therefore, the units digit of \(66^6 - 5^{55}\) will be:
\(\{ smaller \ number \ with \ the \ units \ digit \ of \ 6 \} - \{ larger \ number \ with \ the \ units \ digit \ of \ 5\} = 9\).
For instance, \(16-25=-9\).
Answer: E
mvmarcus7
Joined: 05 Mar 2014
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15 Dec 2019, 07:08
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I think I thought of one alternative approach to "see" this question.
Since 5^55 >> 66^6, we have that 66^6 - 5^55 will result in a negative number.
While we know that the units digit of 66^6 will be 6, and the units digit of 5^55 will be 5, we will have (6 - 5). But we know that the "5" number will be larger than the "6" number, then we will need to get a value from the tens digit.
When we "get" the tens digit, we get a 10,
Then we will have 10 - (6 - 5) = 9.
This is like when we have 95 - 46, and we "borrow" 10 from 9 (which then becomes 8), to turn 5 into 15. This 15 we subtract from 6, to have 9 on the units digit.
While on the tens digit we have 8 - 4 = 4, which connecting the numbers we have 49, 95 - 46 = 49.
Great question by the way
Since 5^55 >> 66^6, we have that 66^6 - 5^55 will result in a negative number.
While we know that the units digit of 66^6 will be 6, and the units digit of 5^55 will be 5, we will have (6 - 5). But we know that the "5" number will be larger than the "6" number, then we will need to get a value from the tens digit.
When we "get" the tens digit, we get a 10,
Then we will have 10 - (6 - 5) = 9.
This is like when we have 95 - 46, and we "borrow" 10 from 9 (which then becomes 8), to turn 5 into 15. This 15 we subtract from 6, to have 9 on the units digit.
While on the tens digit we have 8 - 4 = 4, which connecting the numbers we have 49, 95 - 46 = 49.
Great question by the way
