Answer:
x=-1/4
Step-by-step explanation:
-4/16=16x/16
-1/4=x
Answer:
C
Step-by-step explanation:
Answer: = 126
Step-by-step explanation:
(5+3+6) * 9
14 * 9
= 126
Answer:
The answer is ![x \equiv 763 \:(mod \: 1147)](https://tex.z-dn.net/?f=x%20%5Cequiv%20763%20%5C%3A%28mod%20%5C%3A%201147%29)
Step-by-step explanation:
The following steps will give a solution to the congruence
![x^{329} \equiv 452 \:(mod \:1147)](https://tex.z-dn.net/?f=x%5E%7B329%7D%20%5Cequiv%20452%20%5C%3A%28mod%20%5C%3A1147%29)
1. <em>Compute Euler's Phi function
.</em>
We have
by prime factorization, so that ![\phi(1147)=\phi(31)\cdot \phi(37)=30\cdot 36= 1080](https://tex.z-dn.net/?f=%5Cphi%281147%29%3D%5Cphi%2831%29%5Ccdot%20%5Cphi%2837%29%3D30%5Ccdot%2036%3D%201080)
because
where p is a prime number.
2. <em>Find positive integers u and v that satisfy
.</em>
We know a solution exists, since
, using the Euclidean algorithm allows us to find the solution ![1 = -151\cdot 329 + 46\cdot 1080](https://tex.z-dn.net/?f=1%20%3D%20-151%5Ccdot%20329%20%2B%2046%5Ccdot%201080)
In order to get positive values for <em>u</em> and <em>v</em>, we modify this solution:
and ![v=-46+329=283](https://tex.z-dn.net/?f=v%3D-46%2B329%3D283)
The equation
![329 \cdot 929-1080\cdot 283=1](https://tex.z-dn.net/?f=329%20%5Ccdot%20929-1080%5Ccdot%20283%3D1)
provides the key to solving the original problem.
3. <em>Compute
by successive squaring. The value obtained gives the solution x.</em>
We have
, so ![x \equiv 452^{929}\:(mod \: 1147)](https://tex.z-dn.net/?f=x%20%5Cequiv%20452%5E%7B929%7D%5C%3A%28mod%20%5C%3A%201147%29)
To use this method start by looking at the exponent 929 and represent it as a sum of powers of 2 this is called the binary expansion of 929. To do this, find the largest power of 2 less than your exponent in this case it’s
. Subtract 512 from 929 getting 417. And continue in this manner to get:
![929=2^9+2^8+2^7+2^5+2^0](https://tex.z-dn.net/?f=929%3D2%5E9%2B2%5E8%2B2%5E7%2B2%5E5%2B2%5E0)
Now
![452^{929}\:(mod \: 1147)=452^{2^9+2^8+2^7+2^5+2^0}\:(mod \: 1147)](https://tex.z-dn.net/?f=452%5E%7B929%7D%5C%3A%28mod%20%5C%3A%201147%29%3D452%5E%7B2%5E9%2B2%5E8%2B2%5E7%2B2%5E5%2B2%5E0%7D%5C%3A%28mod%20%5C%3A%201147%29)
So all you have to do is to calculate the numbers
![452^{2^9} \:(mod \:1147),452^{2^8} \:(mod \:1147),452^{2^7} \:(mod \:1147),452^{2^5} \:(mod \:1147), 452^{2^0} \:(mod \:1147)](https://tex.z-dn.net/?f=452%5E%7B2%5E9%7D%20%5C%3A%28mod%20%5C%3A1147%29%2C452%5E%7B2%5E8%7D%20%5C%3A%28mod%20%5C%3A1147%29%2C452%5E%7B2%5E7%7D%20%5C%3A%28mod%20%5C%3A1147%29%2C452%5E%7B2%5E5%7D%20%5C%3A%28mod%20%5C%3A1147%29%2C%20452%5E%7B2%5E0%7D%20%5C%3A%28mod%20%5C%3A1147%29)
and multiply them together, then take the product ![(mod \: 1147)](https://tex.z-dn.net/?f=%28mod%20%5C%3A%201147%29)
![452^{2^9} \:(mod \:1147)=417\\452^{2^8} \:(mod \:1147)=359\\452^{2^7} \:(mod \:1147)=565\\452^{2^5} \:(mod \:1147)=417\\452^{2^0} \:(mod \:1147)=452](https://tex.z-dn.net/?f=452%5E%7B2%5E9%7D%20%5C%3A%28mod%20%5C%3A1147%29%3D417%5C%5C452%5E%7B2%5E8%7D%20%5C%3A%28mod%20%5C%3A1147%29%3D359%5C%5C452%5E%7B2%5E7%7D%20%5C%3A%28mod%20%5C%3A1147%29%3D565%5C%5C452%5E%7B2%5E5%7D%20%5C%3A%28mod%20%5C%3A1147%29%3D417%5C%5C452%5E%7B2%5E0%7D%20%5C%3A%28mod%20%5C%3A1147%29%3D452)
![x \equiv 452^{929}\:(mod \: 1147)\\x \equiv 417 \cdot 359\cdot 565 \cdot 417\cdot 452 \:(mod \: 1147)\\x \equiv 121\cdot 376 \:(mod \: 1147)\\x \equiv 763 \:(mod \: 1147)](https://tex.z-dn.net/?f=x%20%5Cequiv%20452%5E%7B929%7D%5C%3A%28mod%20%5C%3A%201147%29%5C%5Cx%20%5Cequiv%20417%20%5Ccdot%20359%5Ccdot%20565%20%5Ccdot%20417%5Ccdot%20452%20%5C%3A%28mod%20%5C%3A%201147%29%5C%5Cx%20%5Cequiv%20121%5Ccdot%20376%20%5C%3A%28mod%20%5C%3A%201147%29%5C%5Cx%20%5Cequiv%20763%20%5C%3A%28mod%20%5C%3A%201147%29)