a.
By Fermat's little theorem, we have
5 and 7 are both prime, so and . By Euler's theorem, we get
Now we can use the Chinese remainder theorem to solve for . Start with
- Taken mod 5, the second term vanishes and . Multiply by the inverse of 4 mod 5 (4), then by 2.
- Taken mod 7, the first term vanishes and . Multiply by the inverse of 2 mod 7 (4), then by 6.
b.
We have , so by Euler's theorem,
Now, raising both sides of the original congruence to the power of 6 gives
Then multiplying both sides by gives
so that is the inverse of 25 mod 64. To find this inverse, solve for in . Using the Euclidean algorithm, we have
64 = 2*25 + 14
25 = 1*14 + 11
14 = 1*11 + 3
11 = 3*3 + 2
3 = 1*2 + 1
=> 1 = 9*64 - 23*25
so that .
So we know
Squaring both sides of this gives
and multiplying both sides by tells us
Use the Euclidean algorithm to solve for .
64 = 3*17 + 13
17 = 1*13 + 4
13 = 3*4 + 1
=> 1 = 4*64 - 15*17
so that , and so
Answer:
- 3 is greater than or equal to x is greater than or equal to 6
Step-by-step explanation:
Hope this helps!!!!
The equation of a line is:
y = mx + c
The m is the gradient of the line, and the c is the y-intercept of the line
That means that the y-intercept of [y = -4x + 3] is 3
and the y-intercept of [y = -4x + 4] is 4
So the distance between the two y-intercepts is:
4 - 3 = <u>1</u>
Answer:
x = 120 degrees
Step-by-step explanation:
50 + 70 = 120
Answer: $156.86
Step by Step:
136.40 x .15 = 20.46
136.40 + 20.46 = 156.86