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Question
Please look at the image attached.
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Thanks a lot.
Hello Tom
To prove that x(s) is in Z, expand by the binomial. The 1/a terms cancel, and all the rest are integral.
ax_1 = 1 mod m, then 1-ax_1 = km with k integral.
Then (1-ax_1)^s = (km)^s.
Now we calculate ax_s = a(1/a - 1/a(km)^s) = 1 - (km)^s = 1 (mod m^s).
So x_s is the solution required.
Best wishes
vijilant
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Vijilant
Expertise
Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.
Experience
Teacher of math for 50 years
Organizations
ATL
Publications
Journal of mathematics and its applications
Education/Credentials
BSc Hons Liverpool
Awards and Honors
State Scholarship 1955
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I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things.
One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.
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