if do - a, + a2 - . . . (-1)*a, is divisible by 11. 33. Show that the number 7, 176,825,942, 116,027,211 is divisible by 9 but not divisible by 11. 34. Show that the number 9,897,654,527,609,805 is divisible by 99. 35. (Test for Divisibility by 3) Let n be an integer with decimal repre- sentation ajak_] . . . a do. Prove that n is divisible by 3 if and only if a tax + . . . + a, + a is divisible by 3. 36. (Test for Divisibility by 4) Let n be an integer with decimal repre- sentation adj. . . a a,. Prove that n is divisible by 4 if and only if a a, is divisible by 4. 37. Show that no integer of the form 111, 111,111, . . .,111 is prime. 38. Consider an integer n of the form a, 111,111,111,111,111,111, 111,111,126. Find values for a and b such that n is divisible by 99. 39. Suppose n is a positive integer written in the form n = a,3* + a_ 3*-1 + . . . + a,3 + a, where each of the a;'s is 0, 1, or 2 (the base 3 representative of n). Show that n is even if and only if a + ak-1 + . . . + a, + a, is even. 40. Find an analog of the condition given in the previous exercise for characterizing divisibility by 4. 41. In your head, determine (2 . 1075 + 2)100 mod 3 and (10100 + 1)99 mod 3. 42. Determine all ring homomorphisms from @ to Q. 43. Let R and S be commutative rings with unity. If o is a homomor- phism from R onto S and the characteristic of R is nonzero, prove that the characteristic of S divides the characteristic of R. 44. Let R be a commutative ring of prime characteristic p. Show that the Frobenius map x -> > is a ring homomorphism from R to R. 45. Is there a ring homomorphism from the reals to some ring whose kernel is the integers? 46. Show that a homomorphism from a field onto a ring with more than one element must be an isomorphism. 47. Suppose that R and S are commutative rings with unities. Let o be a ring homomorphism from R onto S and let A be an ideal of S. a. If A is prime in S, show that p-'(A) = (x E R | Q(x) E A} is prime in R.if do - a, + a2 - . . . (-1)*a, is divisible by 11. 33. Show that the number 7, 176,825,942, 116,027,211 is divisible by 9 but not divisible by 11. 34. Show that the number 9,897,654,527,609,805 is divisible by 99. 35. (Test for Divisibility by 3) Let n be an integer with decimal repre- sentation ajak_] . . . a do. Prove that n is divisible by 3 if and only if a tax + . . . + a, + a is divisible by 3. 36. (Test for Divisibility by 4) Let n be an integer with decimal repre- sentation adj. . . a a,. Prove that n is divisible by 4 if and only if a a, is divisible by 4. 37. Show that no integer of the form 111, 111,111, . . .,111 is prime. 38. Consider an integer n of the form a, 111,111,111,111,111,111, 111,111,126. Find values for a and b such that n is divisible by 99. 39. Suppose n is a positive integer written in the form n = a,3* + a_ 3*-1 + . . . + a,3 + a, where each of the a;'s is 0, 1, or 2 (the base 3 representative of n). Show that n is even if and only if a + ak-1 + . . . + a, + a, is even. 40. Find an analog of the condition given in the previous exercise for characterizing divisibility by 4. 41. In your head, determine (2 . 1075 + 2)100 mod 3 and (10100 + 1)99 mod 3. 42. Determine all ring homomorphisms from @ to Q. 43. Let R and S be commutative rings with unity. If o is a homomor- phism from R onto S and the characteristic of R is nonzero, prove that the characteristic of S divides the characteristic of R. 44. Let R be a commutative ring of prime characteristic p. Show that the Frobenius map x -> > is a ring homomorphism from R to R. 45. Is there a ring homomorphism from the reals to some ring whose kernel is the integers? 46. Show that a homomorphism from a field onto a ring with more than one element must be an isomorphism. 47. Suppose that R and S are commutative rings with unities. Let o be a ring homomorphism from R onto S and let A be an ideal of S. a. If A is prime in S, show that p-'(A) = (x E R | Q(x) E A} is prime in R