How many fractions are there at least true with a denominator of 143?

In the simplest true fraction, the numerator and denominator are irreducible, then the numerator cannot contain factors 11 and 13.

Since 143 = 11*13, then from 1 to 142, there are 12 numbers with 11 factors, and 10 numbers with 13 factors.

Answer: There are 120 minimum true fractions with a denominator of 143.

143=11x13

There are 143 numbers from 1 to 143, of which there are 13 multiples of 11, 11 multiples of 13, and the last one is the same 143, so the multiple of (13+11-1)=23

So the minimum true fraction with a denominator of 143 is 143-23=120.

Let's go backwards, first calculate how many numbers in 1-143 are not the simplest true fractions, and then subtract them.

143=11*13 means that if the numerator is a multiple of 11 or 13, it is not the simplest true fraction.

143 11=13 means that there are 13 numbers that are multiples of 11.

143 13 = 11 means that there are 11 numbers that are multiples of 13.

Only 143 is a common multiple of 11 and 13.

Explain that 1-143 has 11+13-1=23 multiples of 11 or 13, and these numbers cannot be used as numerators.

The remaining 143-23 = 120 numbers as a numerator can make it the simplest fraction.

So the denominator is 143, and the simplest true fraction has 120.

Hope, thank you.

Because 11*13=143

So the numerator is 1-142 for all numbers except 11 and 13.

Therefore, the sum of the most simple and true fractions = 10129 = 143/149 = 10129

Summary. Hello <>

The denominator is 1463, and there are 1462 fractions in total.

The denominator is 1463.

Hello <>

The denominator is 1463, and there are 1462 fractions in total.

Dear, is there anything else you don't understand?You can tell me about your situation in detail, so that I can give you a more detailed answer.

First of all, the absolute fraction of the simplest true silver segment refers to the ignition difference rational number with a real part of 0 and the common divisor of the numerator and denominator of 1. Therefore, for the simplest true fraction with a denominator of 1463, its numerator can only be a positive integer between 1 and 1462 and 1463 without a common divisor, with a total of 1462 choices. Therefore, there are 1462 fractions with a denominator of 1463.

Sharp posture <>

The denominator is the simplest true fraction of 1463, i.e. a fraction in which the numerator and denominator have no common factor, which can be calculated by enumerating the numerator. The first thing to know is that if the denominator is a fixed value of 1463, the true fraction ranges from 1 1463 to 1462 1463. For every positive integer k less than 1463, if k is coprime with 1463, then k 1463 is the simplest true fraction of 1463.

Thus, the problem can be transformed into finding how many numbers from 1 to 1462 are coprime with 1463. According to the definition of Euler's function, if Latanguo n is a positive integer, and phi(n) represents the number of positive integers coprime with n in 1 to n, then the number of positive integers coprime with 1463 in 1 to 1462 is phi(1463). However, solving this Euler function requires prime factor decomposition of 1463, which is highly complex.

However, we can take advantage of the properties of the Euler function: if n is prime, then phi(n) = n - 1. Since Janetan 1463 is not a prime number, but it has only two prime factors 17 and 863 (which can be obtained by methods such as trial division or pollard-rho algorithm), so phi(1463) =17-1) times(863-1)=14592.

To sum up, the denominator is 1463, and there are 14592 fractions in total.

1463x (1-1 7-1 11-1 19+1 7*11+1 7*19+1 Dusty 11*19+1 Sakura 7*11*19).

1082 pcs.

153 = 3 2 17,153 in the division of the state Pei 3 in the number of sails Shen only times filial piety 51, 17 multiples of 9, plus 3 multiples of 3 and 17 times at the same time.

The simplest true score is: 153-51-9+3=96.

11 pcs. Turning 140 into the simplest fraction, the denominator of the excitation cover is 140, which is divisible by , and 140, so there are 11 simplest true fractions with a denominator of 140.

Because 154 = 2 7 11, the simplest true fraction with a denominator of 154 is the chain year: the numerator cannot be even, and the odd number cannot be a multiple of 7 and 11 154 Among these 153 numbers, there are 76 even numerators, 153 7 = 21 numerators are multiples of 7, and the numerators are multiples of 11.

Solution: The simplest true fraction with a denominator of 15 is 15 and 14 15.

Then 1 15+2 15+4 15+7 15+8 15+11 15+13 15+14 15

The simplest true fraction with a denominator of 15 is:

The sum of them is:

1 15+2 15+3 15+4 15+5 15+6 15+7 15+8 15+9 15+10 15+11 15+12 15+13 15+14 15 Approximate 4.

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