1 236ELEMENTARY NUMBER THEORY11.est known even amicable pair whose sum fails to enjoy this feature is 66603025.and 696630544.12PROBLEMS 11.3= 212 (213 – 1) is1. Prove that the Mersenne number M13 is a prime; hence, the integer n =1perfect.[Hint: Because M13divisors of M13 are 53 and 79.]2. Prove that the Mersenne number M19 is a prime; hence, the integer n = 218(219 – 1) isperfect.[Hint: By Theorems 11.5 and 11.6, the only prime divisors to test are 191, 457, and 647.]3. Prove that the Mersenne number M29 is composite.4. A positive integer n is said to be a deficient number if o (n) < 2n and an abundant numberif o (n) > 2n. Prove each of the following:(a) There are infinitely many deficient numbers.[Hint: Consider the integers n = p*, where p is an odd prime and k > 1.](b) There are infinitely many even abundant numbers.[Hint: Consider the integers n = 2* . 3, where k > 1.](c) There are infinitely many odd abundant numbers.[Hint: Consider the integers n = 945 · k, where k is any positive integer not divisibleby 2, 3, 5, or 7. Because 945 = 33 . 5.7, it follows that gcd(945, k)o(n) = 0 (945)0 (k).]5. Assuming that n is an even perfect number and d |n, where 1 < d

Name: 1 236ELEMENTARY NUMBER THEORY11.est known even amicable pair whose sum
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Description: 1 236ELEMENTARY NUMBER THEORY11.est known even amicable pair whose sum fails to enjoy this feature is 66603025.and 696630544.12PROBLEMS 11.3= 212 (213 – 1) is1. Prove that the Mersenne number M13 is a prime; hence, the integer n =1perfect.[Hint: Beca