6. Find these values of the Euler φ-function.a) φ(4) b) φ(10) c) φ(13)7. What are the greatest common divisors of these pairs of integers?a) 37 · 53 · 73, 211 · 35 · 59 b) 11 · 13 · 17, 29 · 37 · 55 · 73 c) 2331, 2317d) 41 · 43 · 53, 41 · 43 · 53 e) 313 · 517, 212 · 721

1. Determine whether each of these integers is prime.a) 21 b) 29 c) 71 d) 97e) 111 f ) 143 g) 93 h) 1012. Find the prime factorization of each of these integers.a) 39 b) 81 c) 101d) 143 e) 289 f ) 899

1. Does 17 divide each of these numbers?a) 68 b) 84 c) 357 d) 10012. What are the quotient and remainder whena) 19 is divided by 7? b) −111 is divided by 11? c) 789 is divided by 23?d) 1001 is divided by 13? e) 0 is divided by 19? f ) 3 is divided by 5?

Let's consider all integers in the range from 11 to n𝑛 (inclusive).Among all pairs of distinct integers in this range, find the maximum possible greatest common divisor of integers in pair. Formally, find the maximum value of gcd(a,b)gcd(𝑎,𝑏), where 1≤a<b≤n1≤𝑎<𝑏≤𝑛.The greatest common divisor, gcd(a,b)gcd(𝑎,𝑏), of two positive integers a𝑎 and b𝑏 is the biggest integer that is a divisor of both a𝑎 and b𝑏.InputThe first line contains a single integer t𝑡 (1≤t≤1001≤𝑡≤100)  — the number of test cases. The description of the test cases follows.The only line of each test case contains a single integer n𝑛 (2≤n≤1062≤𝑛≤106).OutputFor each test case, output the maximum value of gcd(a,b)gcd(𝑎,𝑏) among all 1≤a<b≤n1≤𝑎<𝑏≤𝑛.

15. Let L be the least common multiple of 173 and 105. Among all of t mon divisors > 1 of 175 and 105, Ict D bc the smallest. Which is correct of t Jowi A) D = 5 and 1050 B) D — 7 and L — 525 C) D = 5 and L = 525 D) D — 7 and L — 1050

Analyze the logical forms of the following statements:(a) 3 is a common divisor of 6, 9, and 15. (Note: You did this in exercise2 of Section 1.1, but you should be able to give a better answer now.)(b) x is divisible by both 2 and 3 but not 4.(c) x and y are natural numbers, and exactly one of them is prime.