Proving Divisibility of Integers: k|mn, k|4m, k|4n

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In summary, the conversation discusses proving the statement "For all positive integers, k, m, n, if k|mn then k|4m or k|4n." The poster attempted to rewrite the divisibility statements using the given equation, but was unsure of how to proceed. However, after trying different arrangements of numbers, they realized that the statement may not be true.
  • #1
notSomebody
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I'm at a loss here. I have no idea how to prove this.

For all positive integers, k, m, n if k|mn then k|4m or k|4n.

Homework Equations


An integer r is divisible by an integer d if and only iff r=ds where s is some integer and d != 0.

The Attempt at a Solution


I tried rewriting the divisibilities.

k|4m
4m = ks

k|4n
4n = kq

k|mn
mn = ky

but I don't know where to go from here.
 
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  • #2
Are you sure you wrote it right? Take k = 21, m = 3, and n = 7 - doesn't work.
 
  • #3
Now I look foolish. I tried a couple of arrangements of numbers and it worked out, so I assumed it to be true. Thanks.
 

FAQ: Proving Divisibility of Integers: k|mn, k|4m, k|4n

How do you prove divisibility of integers?

To prove divisibility of integers, we must show that one integer is a factor of another integer. This can be done by using the definition of divisibility, which states that if a is divisible by b, then there exists an integer c such that a = bc.

What is the divisibility rule for 4?

The divisibility rule for 4 states that a number is divisible by 4 if its last two digits are divisible by 4. In other words, if the number formed by the last two digits is divisible by 4, then the entire number is divisible by 4.

How do you prove that k|mn?

To prove that k|mn, we must show that k is a factor of both m and n. This can be done by using the definition of divisibility and showing that there exists an integer c such that m = kc and n = kc.

Can k be any integer in k|mn, k|4m, k|4n?

Yes, k can be any integer in these statements. As long as k is a factor of both m and n, or 4m and 4n, then the statements are true.

How can you use the divisibility rule for 4 to prove k|4m and k|4n?

To prove k|4m and k|4n, we can use the divisibility rule for 4 and the fact that k is a factor of both m and n. Since k is a factor of m and n, and the last two digits of 4m and 4n are divisible by 4, we can conclude that k is also a factor of 4m and 4n.

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