Modulo operation | What does this mean?

[michonamona]

Homework Statement

Suppose i and j take on values from {0,1,2,...,7}. We say that i and j are 'happy' if i - j (their difference) is congruent to 1.4 or 7 modulo 8.

Note: 'happy' is some mathematical property not relevant to the question.


What does 7 modulo 8 mean? What does it mean for i - j to be congruent to 1.4?

Thank you for your help!

M

 

[Robert1986]

Homework Statement

Suppose i and j take on values from {0,1,2,...,7}. We say that i and j are 'happy' if i - j (their difference) is congruent to 1.4 or 7 modulo 8.

Note: 'happy' is some mathematical property not relevant to the question. What does 7 modulo 8 mean? What does it mean for i - j to be congruent to 1.4?

Thank you for your help!

M

I'm not sure what you mean by i - j congruent to 1.4 modulo 8, did you mean: "...congruent to 1,4 or 7 modulo 8?" Either way, I shall attempt to explain this concept.

Do you remember when you first learned division, and you didn't care about decimal places? That is, if you divided one integer into another and didn't get an even answer, you reported the remainder as a part of your answer, right? For example, 8/3 = 2 R.2, right? Well, it turns out that this is a pretty important thing to study in number theory. So, we say that two numbers, m and k are congruent modulo n when the remainder of n/m is the same as the remainder of n/k. This is equivalent to saying that k-m (or m-k) is an integer multiple of n.

The congruency relation is an equivlancy relation. In this context, this means that for the arithmetic operations, you can treat the congruency relation just like "=". For example, if a is congruent to b mod n and c is congruent to d mod n, the a+b is congruent to c+d mod n.

 

[Mark44]

I'm not sure what you mean by i - j congruent to 1.4 modulo 8, did you mean: "...congruent to 1,4 or 7 modulo 8?"

I'm pretty sure that was a typo somewhere that should read "i - j is congruent to 1, 4, or 7 modulo 8."