How it Works › Forums › Advanced Problems (AMC 8 Problems 20-25 and beyond) › Unique Properties of Integers Advanced Problems Set 2 › Reply To: Unique Properties of Integers Advanced Problems Set 2

Yash Agarwal

#2:

Mod Rules needed for this problem:

If A = B (Mod m), then:

A + C = B + C (Mod m)

A^x = B^x (Mod m)

First, let’s write out the squares of the first 6 positive integers:

1, 4, 9, 16, 25, and 36

So, if x^2 = 2 (mod 7), that means:

x^2 = 9 (mod 7), since the addition rule states that you can add the mod to the number as many times as you like. For example, if a number was x (mod m) then that would be equal to x + m (mod m).

So, we have x^2 = 9 (mod 7)

9 = 3^2

So, we can rewrite this as:

X^2 = 3^2 (mod 7)

So, by the exponential rule, x can equal 3. So, 3 is one of our solutions.

9 + 7 = 16, another one of our perfect squares. So, the other solution is the square root of 16, which is 4. If we keep on adding 7, we would get 23, 30, 37… We have already exceeded 36. So, the only solutions for the first part of this problem are ** 3 and 4. ** Wait, shouldn’t -3 and -4 be involved in the solution too? No, because it has to be a ** positive integer. **

The second part of this problem is “What about x^2 = 1 (mod 15)?”

The rules needed for this problem are:

X – C = Y – C (mod m) if X = Y (mod m)

If x^2 = 1(mod 15), that means that x^2 -1 = 0 (mod 15). In other words, x^2-1/15 leaves a remainder of 0, which means that x^2-1 is divisible by 15. X^2 – 1 = (X+1) (X-1). (X+1) – (X – 1) = 2.

The factors of 15 are 1, 3, 5, and 15. We needed factors that are 2 apart. 3 and 5 is the only pair. 3 + 1 = 4. 5 – 1 = 4. 2 + 1 = 3. 2 – 1 = 1. But, since is not a factor, it is not included here, but it is needed. 0 is a multiple of 13. So, if X-1 = 0, that means X can equal 1 too. ** So, the only solutions for the second part of this problem are 1 and 4.**

This is interesting because 4^2 is not 1. 1 is only 1^2 (and (-1)^2, but they only want positive integers here).

** This response is written by Yash Agarwal. **