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03 Nov 2023, 07:21
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B
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Difficulty:
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53% (02:40) correct
47%
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If n is a positive integer less than or equal to 100, what is the probability that \(1^n + 2^n + 3^n + 4^n\) is not divisible by 5??
A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4
A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4
03 Nov 2023, 08:34
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Bunuel
We can use the concept of cyclicity to solve this question. As we can see for the exponent values between 1 and 4, inclusive, for one value, n = 4, the sum ends in '\(4\)' and hence is not divisible by 5.
Hence, for every four consecutive powers, one power is not divisible by 5.
Required Probability = \(\frac{1}{4}\)
Attachment:
Screenshot 2023-11-03 204229.png [ 8.62 KiB | Viewed 4634 times ]
Option B
07 May 2025, 04:08
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If n is a positive integer less than or equal to 100, what is the probability that \(1^n + 2^n + 3^n + 4^n\) is not divisible by 5?
We can try increasing n and looking for a pattern.
n = 1: 1 + 2 + 3 + 4 = 10 yes
n = 2: 1 + 4 + 9 + 16 = 30 yes
n = 3: 1 + 8 + 27 + 64 = 100 yes
n = 4: 1 + 16 + 81 + 256 =
For this one, we don't have to complete the addition if we note that the tens and hundreds will be divisible by 5. So, all that matters is the units digit of the sum, which is based on the units digits of the numbers.
1 + 6 + 1 + 6 = 14. So, the units digit of the sum is 4, and this sum is not divisible by 5.
n = 5: 1 + (16*2) + (81*3) + (256*4) =
If we notice that the units digits are going to be 1, 2, 3, and 4, we can see that we've found our answer. After all, the units digits determine whether the sum is divisible by 5. So, we've started over at the beginning, and as we increase the powers, the same units digits will repeat.
So, we've found the pattern. As n increases, 3/4 of the sums will be divisible by 5, and 1/4 will not be.
A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4
Correct answer: B
We can try increasing n and looking for a pattern.
n = 1: 1 + 2 + 3 + 4 = 10 yes
n = 2: 1 + 4 + 9 + 16 = 30 yes
n = 3: 1 + 8 + 27 + 64 = 100 yes
n = 4: 1 + 16 + 81 + 256 =
For this one, we don't have to complete the addition if we note that the tens and hundreds will be divisible by 5. So, all that matters is the units digit of the sum, which is based on the units digits of the numbers.
1 + 6 + 1 + 6 = 14. So, the units digit of the sum is 4, and this sum is not divisible by 5.
n = 5: 1 + (16*2) + (81*3) + (256*4) =
If we notice that the units digits are going to be 1, 2, 3, and 4, we can see that we've found our answer. After all, the units digits determine whether the sum is divisible by 5. So, we've started over at the beginning, and as we increase the powers, the same units digits will repeat.
So, we've found the pattern. As n increases, 3/4 of the sums will be divisible by 5, and 1/4 will not be.
A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4
Correct answer: B
