Prove that there does not exist an integer c and a natural number x ≥ 2 such that c ≡ 2 (mod x) and c ≡ 3 (mod x^3 ).

 

By the Chinese Remainder Theorem, the system is solvable if gcd(x,x^3) | 2 -3  which implies x | 1. How do i continue from here?

in Other Math Topics by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

The Chinese Remainder Theorem cannot be used if any modulus shares a factor with the others. In this case x³ and x have factor x in common.

c=ax+2 is the same as c≡2(mod x),

c=bx³+3 is the same as c≡3(mod x³), where a and b are natural numbers.

Therefore:

ax+2=bx³+3=x(bx²+3/x).

Divide through by x:

a+2/x=bx²+3/x.

If x=1 then a+2=b+3 and b=a-1, but we are only interested in x≥2;

If x=2 then a+1=4b+3/2, but we have an integer on the left-hand side and a mixed number on the right-hand side. Equality is not possible.

If x=3 then a+2/3=9b+1. Again, equality is not possible because RHS is an integer but LHS is a mixed number.

For x>3, a and bx² are integers, both LHS and RHS are mixed numbers, but 2/x cannot be equal to 3/x for x>1. This implies that there can be no integer c for which c≡2(mod x) and c≡3(mod x³).

by Top Rated User (839k points)

Related questions

1 answer
1 answer
1 answer
asked Mar 21, 2013 in Algebra 2 Answers by anonymous | 325 views
1 answer
asked Sep 5, 2012 in Algebra 1 Answers by anonymous | 934 views
0 answers
asked Feb 28, 2014 in Pre-Algebra Answers by shagira780 Level 1 User (140 points) | 156 views
1 answer
asked Oct 20, 2016 in Other Math Topics by marya | 912 views
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
86,305 questions
92,362 answers
2,251 comments
23,927 users