5 5. Prove that the integer 53103 +10353 is divisible by 39, and that 111333 +333|11 is divis-4. (a) Find the remainders when 250 and 410 are divided by 7.that c 0 (mod n). Indeed, if c = 0 (mod n), then gcd(c, n) = n and the conclusionLet us call attention to the fact that, in Theorem 4.3, it is unnecessary to stipulate11 = 5 (mod 3). A further illustration is given by the congruence –35 = 45 (mod 8),Corollary 2. If ca = cb (mod p) andp X c, where p is a prime number, thenerd 41).shall haveExample 4.4. Consider the congruence 33 = 15 (mod 9) or, if one prefers,3.11 = 3.5 (mod 9). Because gcd(3, 9) = 3, Theorem 4.3 leads to the conclusion thatwhich is the same as 5 (-7) = 5 ·9 (mod 8). The integers 5 and 8 being relativelyprime, we may cancel the factor 5 to obtain a correct congruence –7 = 9 (mod 8).a = b (mod p).toshe theorem would state that a = b (mod 1); but, as we remarked earlier, thisholds trivially for all integers a and b.There is another curious situation that can arise with congruences: The productof two integers, neither of whích is congruent to zero, may turn out to be congruent tozero. For instance, 4 3 0 (mod 12), but 4 #0 (mod 12) and 3 0 (mod 12). It is asimple matter to show that if ab = 0 (mod n) and gcd(a, n) = 1, then b = 0 (mod n):Corollary 1 permits us legitimately to cancel the factor a from both sides of thecongruence ab = a · 0 (mod n). A variation on this is that when ab = 0 (mod p),pa prime, then either a = 0 (mod p) or b = 0 (mod p).withPROBLEMS 4.21. Prove each of the following assertions:(a) If a = b (mod n) and m |n, then a = b (mod m).(b) If a = b (mod n) and c > 0, then ca = cb (mod cn).(C) If a = b (mod n) and the integers a, b, n are all divisible by d > 0, thena/d = b/d (mod n/d).** Give an example to show that a? = b? (mod n) need not imply that a = b(mod n).3. If a = b (mod n), prove that gcd(a, n) = gcd(b, n).(0) What is the remainder when the following sum is divided by 4 ?15+25 + 35+...+99$ + 100°ible by 7.

Name: 5 5. Prove that the integer 53103 +10353 is divisible by 39, and that
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Description: 5 5. Prove that the integer 53103 +10353 is divisible by 39, and that 111333 +333|11 is divis-4. (a) Find the remainders when 250 and 410 are divided by 7.that c 0 (mod n). Indeed, if c = 0 (mod n), then gcd(c, n) = n and the conclusionLet us call at