4 Corollary 1.Cb (mod n) and gcd(c, n) = 1, then a = b (mod n).We take a moment to record a special case of Corollary 1 that we shall havefrequent occasion to use, namely, Corollary 2.Corollary 2. If ca = cb (mod p) and p X c, where p is a prime number, thena = b (mod p).Proof. The conditions p Xc and p a prime imply that gcd(c, p) = 1.Example 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 that11 = 5 (mod 3). A further illustration is given by the congruence -35 = 45 (mod 8),which 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).Let us call attention to the fact that, in Theorem 4.3, it is unnecessary to stipulatethat c 0 (mod n). Indeed, if c = 0 (mod n), then gcd(c, n) = n and the conclusionof the 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 which 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),with p a prime, then either a = 0 (mod p) or b = 0 (mod p).PROBLEMS 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).2. Give an example to show that a2 = b² (mod n) need not imply that a = b(mod n).3. If a = b (mod n), prove that gcd(a, n) = gcd(b, n).4. (a) Find the remainders when 250 and 4165 are divided by 7.(b) What is the remainder when the following sum is divided by 4?15 + 25 + 35 + ...+99$ + 100$5. Prove that the integer 53103 + 10353 is divisible by 39, and that 111333 +33311 ;s divisible by 7.

Name: 4 Corollary 1.Cb (mod n) and gcd(c, n) = 1, then a = b (mod n).We take
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Description: 4 Corollary 1.Cb (mod n) and gcd(c, n) = 1, then a = b (mod n).We take a moment to record a special case of Corollary 1 that we shall havefrequent occasion to use, namely, Corollary 2.Corollary 2. If ca = cb (mod p) and p X c, where p is a prime numb

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4 Corollary 1.Cb (mod n) and gcd(c, n) = 1, then a = b (mod n).We take a moment to record a special case of Corollary 1 that we shall havefrequent occasion to use, namely, Corollary 2.Corollary 2. If ca = cb (mod p) and p X c, where p is a prime number, thena = b (mod p).Proof. The conditions p Xc and p a prime imply that gcd(c, p) = 1.Example 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 that11 = 5 (mod 3). A further illustration is given by the congruence -35 = 45 (mod 8),which 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).Let us call attention to the fact that, in Theorem 4.3, it is unnecessary to stipulatethat c 0 (mod n). Indeed, if c = 0 (mod n), then gcd(c, n) = n and the conclusionof the 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 which 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),with p a prime, then either a = 0 (mod p) or b = 0 (mod p).PROBLEMS 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).2. Give an example to show that a2 = b² (mod n) need not imply that a = b(mod n).3. If a = b (mod n), prove that gcd(a, n) = gcd(b, n).4. (a) Find the remainders when 250 and 4165 are divided by 7.(b) What is the remainder when the following sum is divided by 4?15 + 25 + 35 + ...+99$ + 100$5. Prove that the integer 53103 + 10353 is divisible by 39, and that 111333 +33311 ;s divisible by 7.

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3. If a = b (mod n), prove that gcd(a, n) = gcd(b, n). 4. (a) Find the remainders when 250 and 4165 are divided by 7. (b) What is the remainder when the following sum is divided by 4? 15 +25 +35+ .. +99° + 1005