If n is a positive integer less than or equal to 100, what is the

Question

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

gmatophobia · Answer

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} Screenshot 2023-11-03 204229.png Option B

MartyMurray · Answer

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

XLmafia21 · Answer

For the expression to be divisible by 5, it should leave remainder as 0.
1^n+2^n+3^n+4^n 
when divided by 5, can be rewritten as 
1^n + 2^n + (-2)^n + (-1)^n
for n = 1 to 100,
for odd 'n', the expression will yield a sum of 0 as the terms will cancel out.
For even 'n',
n = 2, sum of unit digits is 10
n = 4, sum of unit digits is 16
n=6, sum of unit digits is 10 and so on.
So, 25 out of 50 even numbers are divisible by 5.

Total 50 + 25 = 75 /100 = 3/4 are divisible

Hence, 1/4 is not divisible. Option B

Krunaal · Answer

Let's test a case for even 'n', taking n = 2;  1^2 + 2^2 + (-2)^2 + (-1)^2  => 1 + 4 + 4 + 1 =  10 which is divisible by 5 , so remainder is 0 for this.

The only cases in even numbers where  'n' is a multiple of 4 , will lead to a sum which is not divisible by 5.

From 1 to 100, we have 25 multiples of 4.

Therefore, 25/100 = 1/4

Hope it helps.

If n is a positive integer less than or equal to 100, what is the

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If n is a positive integer less than or equal to 100, what is the

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