Theorem: permit a, b, and c it is in integers through a e 0 and also b e 0. If a|b and b|c, then a|c.

In order to prove this statement, we an initial need to know what the mathematics notation colorreda|b implies.

You are watching: If a divides b and b divides c then a divides c

I have a different lesson stating the an interpretation of a|b.

To review, the math notation a|b is review as “a divides b “. The presumption is that both a and also b room integers yet a doesn’t same zero, a e 0. In addition, the vertical bar in a|b is referred to as pipe.


As it stands, the notation a|b is not helpful to us since in its present form, there’s no method that we can algebraically manipulate it. Us must convert it in an equation form.

Here’s the thing, a|b deserve to be created in the equation as b = ar wherein r is an integer.


For example, in 2|10, we recognize that 2 evenly divides 10. That method there is one integer as soon as multiplied to 2 provides a product that 10.

What could that number be? that is colorred5 because 2 imes 5 = 10.

Thus, we say 2|10 means 10 = 2left( 5 ight)



Note: The objective of brainstorming in creating proof is for us to understand what the theorem is trying come convey; and gather sufficient information to connect the dots, which will certainly be offered to leg the hypothesis and also the conclusion.

Since we space using the method of straight proof, we want to display that we have the right to manipulate the hypothesis to come at the conclusion.

Hypothesis: a divides b and b divides c

Conclusion: a divides c


Now, let’s express every notation into an equation. Us hope the by doing for this reason will disclose an chance so we can proceed v our heat of reasoning.

a|bb = am ← Equation #1m is an integer
b|cc = bn ← Equation #2n is an integer

What should we do next? Well, we have the right to substitute the expression because that b the Equation #1 right into the b the Equation #2.


After substitution, we get the one below.

c = left( am ight)n

Apply the Associative property of Multiplication. An alert that the grouping symbol (parenthesis) move from am to mn.

The Associative residential property of Multiplication assures that when multiplying numbers, the product is constantly the same no matter just how we group the numbers. Thus, left( am ight)n = aleft( mn ight).

This property enables us come rewrite the equation there is no breaking any math laws since the 2 equations might look different however they are essentially the exact same or equivalent.

I expect you can see currently why we have to perform such slight adjustment using the Associative Property.

c = left( am ight)n → c = aleft( mn ight)

After we substitute the expression the largeb indigenous Equation #1 right into the largeb that Equation #2, and apply the Associative residential or commercial property of Multiplication, us are ready to move to the next step.

Notice that inside the parenthesis space two arbitrarily integers that space being multiplied.

If friend remember, over there is a basic yet an extremely useful residential property of the set of Integers ( the symbol for the collection of integers is mathbbZ ).

The property is referred to as the Closure residential or commercial property of Multiplication. It states that if m and n space integers then the product of m and n is additionally an integer. Therefore, m imes n in mathbbZ.

From where we left off, we have actually

c = aleft( mn ight).

Since mn is just an additional integer using the Closure residential property of Multiplication, that way we have the right to let mn = k wherein k is one integer.

We deserve to rewrite c = aleft( mn ight) together c = aleft( k ight).

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The equation c = aleft( k ight) have the right to be expressed in notation form as a|c which way that a divides c.

This is exactly where we desire to show! currently it’s time to create the yes, really proof.


THEOREM: allow a, b, and also c it is in integers v a e 0 and also b e 0. If a|b and also b|c, climate a|c.

PROOF: intend a, b, and c room integers where both a and b carry out not same to zero. Due to the fact that a divides b, a|b, climate there exist an creature m such the b = to be (Equation #1). Similarly, because b divides c, b|c, there exists an essence n such that c=bn (Equation #2). Now, instead of the expression of b indigenous Equation #1 into the b in Equation #2. By doing so, the equation c=bm is revolutionized to c=(am)n. Next, use the Associative residential property of Multiplication ~ above the equation c=(am)n to acquire c=a(mn). Due to the fact that m and also n space integers, your product must additionally be an creature by the Closure building of Multiplication; the is, m imes n in mathbbZ. Let k = m imes n. In the equation c=a(mn), instead of mn through k to attain c=ak.The equation c=ak implies that a divides c or when written in shorthand we have a|c. Therefore, we have actually proved the a divides c. ◾️