Decide whether each of these integers is congruent to 3 modulo 7.a) 37________________b) 66________________c) -17________________d) -67

Solution:Step 1Here, in this problem we have to calculate whether each of the following integer is congruent to 3 modulo 7.We can say an integer a is congruent to b modulo m, if a and b are integers and m is a positive integer. And also m divides( a-b) completely.Or, value of .We use mathematical notation to state that integer a is congruent to b modulo m. If integer a-b is not divisible by m, then we can say integer a is not congruent to b modulo m, Step 2a) 37We can write So, here we have 3 modulo 7 = 3Now, we have to calculate 37 modulo 7We can write So, here we have 37 modulo 7 =2Since, Therefore integer 37 is not congruent to 3 modulo 7.Or we can also check according to definition that ) integer 34 (a-b) is not divisible by m=7.Therefore integer 37 is not congruent to 3 modulo 7.Step 3b) 66We can write So, here we have 3 modulo 7 = 3Now, we have to calculate 66 modulo 7We can...