Find the remainder of 2^100/12
A. 4
B. 2
C. 8
D. 1
E. none
The OA is the option A.
Is there a simple way to solve this PS question? I can I solve it? I'd appreciate your help.
Find the remainder of 2^100/12
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Test easy exponents and look for a pattern.Vincen wrote:Find the remainder of 2^100/12
A. 4
B. 2
C. 8
D. 1
E. none
2²/12 = 4/12 = 0 R4.
2³/12 = 8/12 = 0 R8.
2�/12 = 16/12 = 1 R4.
2�/12 = 32/12 = 2 R8.
2�/12 = 64/12 = 5 R4.
2�/12 = 128/12 = 10 R8.
The cases above illustrate the following:
When the exponent is EVEN, the remainder is 4.
When the exponent is ODD, the remainder is 8.
Since 2¹��/12 has an even exponent, the remainder will be 4.
The correct answer is A.
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Let's find a remainder pattern:Vincen wrote:Find the remainder of 2^100/12
A. 4
B. 2
C. 8
D. 1
E. none
2^1/12 = 0 remainder 2
2^2/12 = 0 remainder 4
2^3/12 = 0 remainder 8
2^4/12 = 1 remainder 4
2^5/12 = 2 remainder 8
2^6/12 = 5 remainder 4
We see that, besides 2^1/12, we have this pattern: when 2 is raised to an even power, the remainder is 4, and when 2 is raised to an odd power the remainder is 8. Thus, 2^100/12 has a reminder of 4.
Answer: A
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Hi Vincen,Find the remainder of 2^100/12
A. 4
B. 2
C. 8
D. 1
E. none
The OA is the option A.
Is there a simple way to solve this PS question? I can I solve it? I'd appreciate your help.
Let's take a look at your question.
We need to find the remainder of
$$\frac{2^{100}}{12}$$
$$=\frac{2^{2\times50}}{12}$$
$$=\frac{4^{50}}{12}$$
Let's check what happens when powers of 4 is divided by 12, starting from 4^2.
$$4^2=16$$
$$\frac{16}{12}=1\ remainder\ 4$$
$$4^3=64$$
$$\frac{64}{12}=5\ remainder\ 4$$
$$4^4=256$$
$$\frac{256}{12}=21\ remainder\ 4$$
$$4^5=1024$$
$$\frac{1024}{12}=85\ remainder\ 4$$
Which shows that remainder is 4 for all the positive powers of 4.
Therefore, option A is correct.
Hope it helps.
I am available if you'd like any follow up.
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