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Question#1: ( 9, 37), The requirements for the value of x is that the number has to leave a remainder of 1 when divided by four, and a remainder of 2 when divided by 7. By the statements I knew that the value had to be greater than 7, so the first answer I got was 9. I decided to find the other one by creating an equation: 9*4+1= 37. If you do the same thing for the divisor 7, you get 65, but the number has to be less than 56, so there are only two answers.

Question#2: (No solution), 4 and 8 come in the same table, so any number can’t equal to the remainders of both.

Question#3: (2), There are 96 numbers, and 48 pairs. Each pair equals -1, so S= -48/10. -48 mod 10= 2.

Question#4: (5, 13, 19, 31, 43) Actually it is true that two composite numbers do equal prime numbers, here are some examples, (25+6=31), (20+21=41), (25+12=37), (15+4=19), (39+8=47), (39+4=43), and many more. The solutions shown above subtracted by 2, and then added by that number equals to the numbers.