Modulus Operations Homework: Simplifying 1^3+2^3+3^3+...+99^3+100^3(mod4)

[Elruso]

Homework Statement

How do you simplify : 1^3+2^3+3^3+4^3+...+99^3+100^3(mod4)

Please try to explain the solution as detailed as possible or atleast so I can understand it.

 

[D H]

Use the fact that (4a+b)^3 mod 4 = b^3 mod 4. All you have to worry about are b=1,2,3,4. All of the other terms in the sum are duplicates.

 

[Architeuthis Dux]

I would approach this problem by first understanding the concept of modulus operations. Modulus operations, denoted by "mod", is a mathematical operation that finds the remainder after division of one number by another. For example, 10 mod 4 = 2, because when we divide 10 by 4, the remainder is 2.

Now, let's apply this concept to the given expression. We have 1^3+2^3+3^3+...+99^3+100^3(mod4). This means we need to find the remainder when this expression is divided by 4.

To simplify this expression, we can first break it down into smaller parts. For example, let's consider the first few terms: 1^3+2^3+3^3+4^3(mod4). We can rewrite this expression as (1^3(mod4) + 2^3(mod4) + 3^3(mod4) + 4^3(mod4))(mod4). This is because the modulus operation can be applied to each individual term separately.

Now, let's find the remainder for each of these individual terms. 1^3(mod4) = 1, 2^3(mod4) = 0, 3^3(mod4) = 3, 4^3(mod4) = 0. This is because when we divide 1 by 4, the remainder is 1. Similarly, when we divide 2 by 4, the remainder is 0. When we divide 3 by 4, the remainder is 3 and when we divide 4 by 4, the remainder is 0.

Now, let's substitute these values back into our expression. We get (1+0+3+0)(mod4). This simplifies to 4(mod4). And finally, the remainder when 4 is divided by 4 is 0.

We can now generalize this approach for the entire expression. Each term in the expression will either have a remainder of 0 or 1 or 2 or 3 when divided by 4. So, we can rewrite the expression as (0+1+2+3+0+1+2+3+...+0+1+2+3)(mod4). This is because we are essentially adding the remainders for each individual term.

Now, we can