Show that 21 x 182n + 36 x 73n is divisible by 19 for all positive integers n.

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mlehuzzah's profile pic

mlehuzzah | Student, Graduate | (Level 1) Associate Educator

Posted on




As the first responder wrote, there is a typo in the problem.

But here is how to solve a problem like this.

If we only care whether a number is divisible by 19, then we want to work with numbers "modulo 19"


What that means is, we are going to consider all of the following numbers to be the same:

`-38 -= -19 -= 0 -= 19 -= 38 -= ...`

 

We are also going to consider all of these to be the same:


`-37 -= -18 -= 1 -= 20 -= 39 -= ...`

 

And all of these as well:


`-36 -= -17 -= 2 -= 21 -= 40 -= ...`


One way to "simplify" a number using modular arithmetic, is to look at the remainder when you divide by 19.

`182 -: 19 = 9 "R" 11`

Thus `182 -= 11 "mod" 19`

or, if it is more convenient, subtract 19 from the remainder, and we also have:

Thus `182 -= -8 "mod" 19`


(So, if a number is divisibly by 19, that is another way of saying it is congruent (`-=`) modulo 19





When you add, subtract, or multiply two numbers, you may first simplify them:

So

`38+21 -= 0 + 2 -= 2 "mod" 19`

or

`21*182 -= 2*11 -= 22 -= 3 "mod" 19`





So, using modular arithmetic, we have that:

`21*182n+36*73n -= 2*11n+(-2)*(-3)n -= 22n+6n -= 28n "mod" 19`


We want 28n to be congruent to 0.  But the only way to make that happen is if 28n is a multiple of 19
That only happens if n=19, 38, ... etc


Here is where the typo might have been:

`21*187n + 36*73n`

Then:

`21 -= 2 "mod" 19`

`187 -= -3 "mod" 19`

`36 -= -2 "mod" 19`

`73 -= -3 "mod" 19`

`21*187n + 36*73n -= (2)(-3)n+(-2)(-3)n -= -6n+6n -= 0 "mod" 19`


So for any n, `21*187n + 36*73n` is divisible by 19

 

 

PS: for more on the subject of modular arithmetic:

http://en.wikipedia.org/wiki/Modular_arithmetic

jeew-m's profile pic

jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted on

when n=1

21*182n+36*73n = 21*182+36*73 = 6450

Then 6450/19 = 339.47 ; This is not a divisible by 19.

 

In the same manner if you put n=2 you will get 12900 as the addition.

Then 12900/19 =678.94 ; This is not a divisible by 19.

When n=1 and n=2 the statement is not correct because they are not divisible by 19. So we cannot say for all integers of n; 21*182n+36*73n is divisible by 19.

There may be some thing wrong in the question. May be a typing error.

 

hghppotcboho's profile pic

hghppotcboho | Student, Grade 9 | eNotes Newbie

Posted on

The problem is meant to be:

show that 21 x 18^2n + 36 x 7^3n is divisable by 19 for all Positive integers n

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