In May 2024
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If A is a finite set having n elements, prove that A has exactly
2n distinct subsets
Prove that there are infinitely many primes
Let "p_1,p_2,\\dots,p_n" be distinct pisitive primes. Show that "(p_1p_2\\dots p_n)+1" is divisible by none of these primes.
Show that if a nd b are positive untegers, then ab=LCM(a,b)*GCD(a,b)
If there are integers a, b, s, and t such that, the sum at+bs=1, show that GCD(a,b)=1
Let "a" and "b" be two integers. If "a|b" and "b|a", them show that "a=\\pm b"
Prove that a ring with one consists only of zero if and only if 1 = 0
Some of the following statements below are true, and some are false. Prove all those statements are true, and for those are false explanation why the statement is false.
a. If R is an integral domain and about=av with a is not equal to 0 then b=c.
b. If F is a field then it must be an integral domain.
c. If R is an integral domain then it must be a field.
d. 4Z is an ideal of (Q,+,•), the ring of rationals.
e. 4Z is a subring of (Q,+,•), the ring of rationals.
Prove the following statement:
Every finite integral domain is a field.
Show directly that 7Z is a maximal ideal of (Z,+,•).
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#340153 on Dec 2023