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10 Nov 2022, 07:46
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D
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Difficulty:
45%
(medium)
Question Stats:
68% (01:37) correct
32%
(01:39)
wrong
based on 161
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For which of the following values of x is the units digit of the product 2*3^x equal to 4?
A. 12
B. 13
C. 14
D. 15
E. 16
A. 12
B. 13
C. 14
D. 15
E. 16
10 Nov 2022, 08:53
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We want the unit digit of \(2*3^x\) equal to 4
The unit digit of 3^x can be one of the following -
The unit digit of \(3^x\) * 2 can be 4 only when the \(3^x\) ends in 7.
Hence x should be of the form 4*z + 3.
Let's evaluate the given options -
A. 12 : 4*3 + 0
B. 13 : 4*3 + 1
C. 14 : 4*3 + 2
D. 15 : 4*3 + 3
E. 16 : 4*4 + 0
IMO D
The unit digit of 3^x can be one of the following -
- 3
- 9
- 7
- 1
The unit digit of \(3^x\) * 2 can be 4 only when the \(3^x\) ends in 7.
Hence x should be of the form 4*z + 3.
Let's evaluate the given options -
A. 12 : 4*3 + 0
B. 13 : 4*3 + 1
C. 14 : 4*3 + 2
D. 15 : 4*3 + 3
E. 16 : 4*4 + 0
IMO D
30 May 2025, 08:33
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↧↧↧ Detailed Video Solution to the Problem ↧↧↧
We need to find the values of x for which the units digit of 2*\(3^x\) = 4
When we multiply some number with 2 then we can get units digit of 4 only when we have the units digit of the number as 2 or 7
As 2*2 = 4 and 2*7 = 14 making the units' digit as 4 in both the cases
=> Units digit of \(3^x\) = 2 or 7
Since 3 is an odd number of units' digit of any power of 3 cannot be 2
=> Units digit of \(3^x\) = 7
Now lets start by finding the cyclicity of units' digit in powers of 3
\(3^1\) units’ digit is 3
\(3^2\) units’ digit is 9
\(3^3\) units’ digit is 7
\(3^4\) units’ digit is 1
\(3^5\) units’ digit is 3
That means that units digit of power of 3 has a cycle of 4
=> In order to get a units digit of 7, we need to divided the power of 3 by 4 to get a remainder of 3
A. 12 , 12 when divided by 4 gives 0 remainder => NOT POSSIBLE
B. 13 , 12 when divided by 4 gives 1 remainder => NOT POSSIBLE
C. 14 , 12 when divided by 4 gives 2 remainder => NOT POSSIBLE
D. 15 , 12 when divided by 4 gives 3 remainder => POSSIBLE
E. 16 , 12 when divided by 4 gives 0 remainder => NOT POSSIBLE
So, Answer will be D
Hope it helps!
Link to Theory for Last Two digits of exponents here
Link to Theory for Units' digit of exponents here
We need to find the values of x for which the units digit of 2*\(3^x\) = 4
When we multiply some number with 2 then we can get units digit of 4 only when we have the units digit of the number as 2 or 7
As 2*2 = 4 and 2*7 = 14 making the units' digit as 4 in both the cases
=> Units digit of \(3^x\) = 2 or 7
Since 3 is an odd number of units' digit of any power of 3 cannot be 2
=> Units digit of \(3^x\) = 7
Now lets start by finding the cyclicity of units' digit in powers of 3
\(3^1\) units’ digit is 3
\(3^2\) units’ digit is 9
\(3^3\) units’ digit is 7
\(3^4\) units’ digit is 1
\(3^5\) units’ digit is 3
That means that units digit of power of 3 has a cycle of 4
=> In order to get a units digit of 7, we need to divided the power of 3 by 4 to get a remainder of 3
A. 12 , 12 when divided by 4 gives 0 remainder => NOT POSSIBLE
B. 13 , 12 when divided by 4 gives 1 remainder => NOT POSSIBLE
C. 14 , 12 when divided by 4 gives 2 remainder => NOT POSSIBLE
D. 15 , 12 when divided by 4 gives 3 remainder => POSSIBLE
E. 16 , 12 when divided by 4 gives 0 remainder => NOT POSSIBLE
So, Answer will be D
Hope it helps!
Link to Theory for Last Two digits of exponents here
Link to Theory for Units' digit of exponents here
