Question 1099922
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(1)   if 41x is a multiple of 11, where x is a digit what is the value of x
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<pre>
The <U>divisibility by 11 rule</U> says 


    An integer number is divisible by  11  if and only if the alternate sum of its digits is divisible by  11.

    See my lesson  <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-11-rule.lesson>Divisibility by 11 rule</A>  in this site.



So, in order for the 3-digit number 41x, where x is a missed digit (or a "hidden" digit),  was divisible by 11, the alternate sum 

    4 - 1 + x

must be multiple of 11.


It is clear that we must (and can) to consider the only case when 

   4 - 1 + x = 11,

which gives  x = 11 - 4 + 1 = 8.


Indeed, the number 418 is a multiple of 11:  418 = 11*38.
</pre>

Solved.
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(2) &nbsp;&nbsp;if 3y5 is a multiple of 11 , where y is a digit what is the value of y.


<pre>
Solve it by the same way.


The equation  3 - y + 5 = 0  gives  y = 3 + 5 = 8.


Indeed, the number 385 is a multiple of 11:  385 = 11*35.


Here I used  0  as a unique appropriate multiple of 11.
</pre>

Solved.
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(3) &nbsp;&nbsp;if 41z2 is a multiple of 6, where z is a digit what is the value of z.


<pre>
O-o-o, I finally got that z is the "tens" digit in this 4-digit number.


OK,  then we need to apply the divisibility rules for 3 and 2.

The "divisibility by 2 rule" is just satisfied, since the last digit of the number ("ones" digit) is even.


The "divisibility by 3 rule" requires the sum of the digits is multiple of 3:

    4 + 1 + z + 2  is divisible by 3

or, which is the same,  7+z must be divisible by 3.

So, the sum 7+z must be 9, or 12, or 15, which gives the possibilities for z to be equal 2, 5, 8.


Let us check three numbers  4122, 4152 and 4182 for divisibility by 6.


The answer is:  The numbers  4122,  4152 and  4182  all are multiples of 6.
</pre>

Solved.



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On divisibility rules for &nbsp;2, &nbsp;3 &nbsp;and &nbsp;6 &nbsp;read in my lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-2-rule.lesson>Divisibility by 2 rule</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-3-rule.lesson>Divisibility by 3 rule</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-6-rule.lesson>Divisibility by 6 rule</A> 

in this site.



For other problems closely related to your in this post, see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Restore-the-omitted-digit-in-a-number-in-a-way-the-number-is-divisible-by-9.lesson>Restore the omitted digit in a number in a way that the number is divisible by 9</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Restore-the-omitted-digit-in-a-number-in-a-way-the-number-is-divisible-by-11.lesson>Restore the omitted digit in a number in a way that the number is divisible by 11</A>.

in this site.