Proof: Divisibility of Integers by 4

In summary: Note that the first term is divisible by ##4## since ##10^k\equiv 0 \pmod 4## for ##k\geq 2##. Hence, ##4## divides ##N## if and only if ##4## divides the number formed by the tens and units digits, which is ##a_1 10 + a_0##.In summary, an integer is divisible by ##4## if and only if the number formed by its tens and units digits is divisible by ##4##. This is true for both positive and negative integers.
  • #1
Math100
782
220
Homework Statement
Establish the following divisibility criteria:
An integer is divisible by ## 4 ## if and only if the number formed by its tens and units digits is divisible by ## 4 ##.
[Hint: ## 10^{k}\equiv 0\pmod {4} ## for ## k\geq 2 ##.]
Relevant Equations
None.
Proof:

Let ## N ## be an integer.
Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##.
Note that ## 10^{k}\equiv 0\pmod {4} ## for ## k\geq 2 ##.
Thus ## 4\mid N\Leftrightarrow N\equiv 0\pmod {4}\Leftrightarrow a_{1}10+a_{0}\equiv 0\pmod {4} ##.
Therefore, an integer is divisible by ## 4 ## if and only if the number formed by its tens and units digits is divisible by ## 4 ##.
 
  • Like
Likes Delta2
Physics news on Phys.org
  • #2
Correct, but why do you say integers and then restrict everything to positive numbers? The same is true for negative numbers.
 
  • Like
Likes Math100
  • #3
fresh_42 said:
Correct, but why do you say integers and then restrict everything to positive numbers? The same is true for negative numbers.
I guess I was a bit thoughtless. So should I mention "Let N be a natural number.", instead?
 
  • Like
Likes Delta2
  • #4
Is this exercise necessarily in "modular arithmetics", or can we solve it like in 5th grade (that is when kids are about 11 yo)?
 
  • #5
Math100 said:
I guess I was a bit thoughtless. So should I mention "Let N be a natural number.", instead?
No. Integers are fine. Everything remains true if you put a minus sign in front of ##N.##

E.g., you can say: Let ##N## be an integer. We assume w.l.o.g. ##N\geq 0## because the negative case can be treated the same way.

w.l.o.g. stands for: without loss of generality. It means that the assumption is allowed because it is no real restriction.
 
  • Like
Likes Delta2 and Math100
  • #6
Math100 said:
Proof:

Let ## N ## be an integer.
Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##.
You can write ##N## as:

##\displaystyle N=\left(a_{m}10^{m-2}+a_{m-1}10^{m-3}+\dotsb + a_{3}10+a_{2}\right) \,100 + a_{1}10+a_{0} ##
 

FAQ: Proof: Divisibility of Integers by 4

What is the definition of divisibility?

Divisibility is the property of an integer being evenly divisible by another integer, with no remainder.

How do you prove that an integer is divisible by 4?

To prove that an integer is divisible by 4, you can use the rule that an integer is divisible by 4 if the last two digits of the number are divisible by 4. Alternatively, you can divide the number by 4 and check if the remainder is 0.

What is the significance of proving divisibility by 4?

Proving divisibility by 4 is important in many mathematical and scientific calculations, as it allows for simplification and easier manipulation of numbers. It is also a fundamental concept in number theory.

Can you provide an example of a proof for divisibility by 4?

Yes, for example, let's prove that 324 is divisible by 4. The last two digits of 324 are 24, which is divisible by 4. Therefore, 324 is divisible by 4.

How is the proof for divisibility by 4 related to other divisibility rules?

The proof for divisibility by 4 is similar to the proof for divisibility by 2, as both rules involve checking the last digit(s) of the number. It is also related to the proof for divisibility by 8, as a number that is divisible by 4 is also divisible by 8.

Similar threads

Replies
2
Views
1K
Replies
7
Views
1K
Replies
3
Views
2K
Replies
6
Views
1K
Replies
1
Views
1K
Replies
3
Views
1K
Replies
1
Views
913
Replies
1
Views
847
Back
Top