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# M23-35

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Math Expert
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A tricky one !!! The answer is option B. Key learning - read each and every word with utmost care !!!
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I think this is a high-quality question and I agree with explanation.
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Hi Bunuel,

Great explanation and a great question. Thanks.

Just for the sake of understanding, could you please let me know what would be the answer to the question if we were given x = 36 instead of x = prime number?

To calculate the number of positive integers less than 36 which do not have a common factor with 36 other than 1 we can do manual counting, in which case I got 11 numbers (5 7 11 13 17 19 23 25 29 31 35).

Could you validate this and let me know if there is any other (a sophisticated) approach?

Thanks again!

Bunuel wrote:
Official Solution:

If $$x$$ is a positive integer, $$f(x)$$ is defined as the number of positive integers which are less than $$x$$ and do not have a common factor with $$x$$ other than 1. If $$x$$ is prime, then $$f(x)=$$?

A. $$x - 2$$
B. $$x - 1$$
C. $$\frac{(x + 1)}{2}$$
D. $$\frac{(x - 1)}{2}$$
E. 2

Basically the question is: how many positive integers are less than given prime number $$x$$, which has no common factor with $$x$$ except 1.

Well, as $$x$$ is a prime, all positive numbers less than $$x$$ have no common factors with $$x$$ (except common factor 1). So there would be $$x-1$$ such numbers (as we are looking number of integers less than $$x$$).

For example consider $$x=7=prime$$: how many numbers are less than 7 and have no common factors with 7: 1, 2, 3, 4, 5, and 6, so total of $$7-1=6$$ numbers.

Math Expert
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theeliteguy wrote:
Hi Bunuel,

Great explanation and a great question. Thanks.

Just for the sake of understanding, could you please let me know what would be the answer to the question if we were given x = 36 instead of x = prime number?

To calculate the number of positive integers less than 36 which do not have a common factor with 36 other than 1 we can do manual counting, in which case I got 11 numbers (5 7 11 13 17 19 23 25 29 31 35).

Could you validate this and let me know if there is any other (a sophisticated) approach?

Thanks again!

Bunuel wrote:
Official Solution:

If $$x$$ is a positive integer, $$f(x)$$ is defined as the number of positive integers which are less than $$x$$ and do not have a common factor with $$x$$ other than 1. If $$x$$ is prime, then $$f(x)=$$?

A. $$x - 2$$
B. $$x - 1$$
C. $$\frac{(x + 1)}{2}$$
D. $$\frac{(x - 1)}{2}$$
E. 2

Basically the question is: how many positive integers are less than given prime number $$x$$, which has no common factor with $$x$$ except 1.

Well, as $$x$$ is a prime, all positive numbers less than $$x$$ have no common factors with $$x$$ (except common factor 1). So there would be $$x-1$$ such numbers (as we are looking number of integers less than $$x$$).

For example consider $$x=7=prime$$: how many numbers are less than 7 and have no common factors with 7: 1, 2, 3, 4, 5, and 6, so total of $$7-1=6$$ numbers.

You missed 1. 1 also does not have have a common factor with 36 other than 1. Otherwise correct.
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Hi Bunuel, can you explain to me what would have happened if you used 9 in your example as opposed to 7? Thanks.
Math Expert
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RatedRKO4 wrote:
Hi Bunuel, can you explain to me what would have happened if you used 9 in your example as opposed to 7? Thanks.

We are told that x is prime, so x cannot be 9. If we were not given that x is prime, then f(9) = 6, there are 6 positive integers which are less than 9 and do not have a common factor with 9 other than 1: 8, 7, 5, 4, 2, and 1.
Math Expert
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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