1,2, and 6 only. Comes from the textbook, Elementary Number Theory 6th edition. Thank you! Many of the next problems can be done by using remainders or congruency, either way is fine with me.1. Let a, b and c integers. Prove or disprove the next statements:a. 7|a" – ab. If 7|a? + b? then 7|a and 7|bc. If 7|a? + b3 + c³ then 7|abc2. Prove that there is no integer n such that 7|4n2 – 3.3. Find the remainder by division by 7 of 99999999.4. Find the remainder in the division by 17 of 15!5. Let p be a prime. Show that for all integer a we have that p|a® + (p – 1)! a andpla + (p – 1)! aPNext some problems not directly related with congruencies but typical of arithmetic.6. Five men collected several coconuts from a deserted island, and they decided to divide thecoconuts the next day. During the night, one of them decided to separate his part so he dividedthe total in 5 groups and give the left-over coconut to a nearby monkey, the man went to sleep.Next, another man did the same with the remainders coconuts and give the left-over coconut tothe monkey. During the night, every man did the same. Next morning the men divide theremainder coconuts and give the left-over coconut to the monkey. Question: What is theminimum number of original coconuts?7. Ihave weights that weight power of 2 kilograms, like 1, 2, 4, 8, 16,..kilograms. Which weightsare necessary to weight 213 kilograms?A slightly more advance ideas.8. As we know the equation ax = 1 (mod n) can be solve if and only if (a, n) = 1, so a and n arerelatively prime. So, knowing the elements relatively prime to another is somewhat important.Let us call (n) the number of positive elements less than n that are relatively prime with n.Prove the next claims about p.a. If p is prime, what is p (p)?b. If p is prime, what is o (p")?c. Ifp and q are prime, what is o(pq)?

Name: 1,2, and 6 only. Comes from the textbook, Elementary Number Theory 6th
Uploaded: 2024-03-05T11:52:21.000Z
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Description: 1,2, and 6 only. Comes from the textbook, Elementary Number Theory 6th edition. Thank you! Many of the next problems can be done by using remainders or congruency, either way is fine with me.1. Let a, b and c integers. Prove or disprove the next stat