How it Works › Forums › Introductory Problems (AMC 8 Problems 10-20) › Unique Properties of Integers Introductory Problems Set 2 › Reply To: Unique Properties of Integers Introductory Problems Set 2

1. First, we can try to find a value that satisfies this condition. We can add 4 to 1 a few times and see if it leaves a remainder of 2 when divided by 7.

1,5 don’t work, but 9 works. After seeing that 9 works, we know that we can just add 28, and the resulting value will have the same remainders since 28 mod 4 =0, 28 mod 7 = 0. So, we get 9,37,65. It asks for values below 56, so there are 2.

2.Since we are told that the number is 0 mod 4, that means that it must be a multiple of 4. However, when any multiple of 4 is divided by 8, the only possible remainders are 0 or 4, so there are none that satisfy this condition.

3.We can group all of these into pairs, and see that there are 96/2 = 48 pairs. Each pair equals -1, so the total is -48 We divide this by 10 to get a remainder of -8 = 2mod 10, so the remainder is 2.