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86% (01:38) correct
14%
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What is the units digit of \(2^{105}+3^{105}+4^{105}+5^{105}?\)
A. 3
B. 4
C. 5
D. 6
E. 7
A. 3
B. 4
C. 5
D. 6
E. 7
22 Mar 2023, 10:13
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What is the units digit of 2^105+3^105+4^105+5^105?
We know that 2 and 3 have cyclicity of 4
so 105/4 gives remainder of 1. Hence, 2^305 has 2 at units place and 3^105 has 3 at its units place
4 has cyclicity of 2 (105/2 gives 1 as remainder, hence, 4^105 has 4 at its units place)and
5 has cyclicity of 1 so 5^105 has 5 at its units place
therefore, sum is 2 + 3 + 4 + 5 = 14
i.e., 4 is units digit.
Answer B
We know that 2 and 3 have cyclicity of 4
so 105/4 gives remainder of 1. Hence, 2^305 has 2 at units place and 3^105 has 3 at its units place
4 has cyclicity of 2 (105/2 gives 1 as remainder, hence, 4^105 has 4 at its units place)and
5 has cyclicity of 1 so 5^105 has 5 at its units place
therefore, sum is 2 + 3 + 4 + 5 = 14
i.e., 4 is units digit.
Answer B
sherifY
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01 Jul 2024, 13:40
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Hello i am a bit confused is there no easier way to do this than this idea of cyclic powers?
I had to find out all the powers in the exam and i dont think thats the fastest way
I had to find out all the powers in the exam and i dont think thats the fastest way
