Mathematics, 06.10.2019 00:10 SweetBriar

How many positive integers satisfy 127=7 (mod n)? n=1 is allowed.

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127=7 (mod n) means when 127 is divided by n, the division leaves a remainder of 7.

The statement is equivalent to
120=0 (mod n), meaning that n divides 120.

All divisors of 120 will satisfy the statement because 120 divided by a divisor (factor) will leave a remainder of 0.

Factors of 120 are:
n={1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}, |n|=16.
You can count how many such values of n there are, and try to check that each one satisfies 127=7 mod n.

Answer from: Quest

let x and 400 - x be the two numbers.

==> 0.80x + 0.85(400 - x) = 400 - 68 from the statement of the problem.

< ==> 0.80x + 0.85(400 - x) = 400 - 68

==> 340 - 0.05x = 332

< ==> -0.05x = -8 ==> x = 160, the first number ==> 400 - x = 240, the second number.

Answer from: Quest

Answer: f(x) = -2x^3 +3x^2 +11x -6see attachedan infinite number. since the magnitude of the leading coefficient is not specified, it may be any negative number. (we have chosen the smallest magnitude integer that makes all coefficients be integers.)step-by-step explanation:

1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.

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How many positive integers satisfy 127=7 (mod n)? n=1 is allowed....

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