Question 1099922: (1)-if 41x is a multiple of 11,where x is a digit what is the value of x
(2)-if 3y5 is a multiple of 11,where y is a digit what is the value of y
(3)-if 41z2 is a multiple of 6,where z is a digit what is the value of z
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
(1)-if 41x is a multiple of 11,where x is a digit what is the value of x
If 41x is a multiple of 11, then x must be a multiple of 11, because 41 is not a multiple of 11. But there is no digit that is a multiple of 11; unless you allow the trivial case of x=0.
(2)-if 3y5 is a multiple of 11,where y is a digit what is the value of y
I assume this is supposed to be 3y^5. In that case the answer is the same as for the first problem. 3 is not a multiple of 11, so y^5 must be a multiple of 11, which means y must be a multiple of 11. Again the only digit for which this is true is the trivial case of y=0.
(3)-if 41z2 is a multiple of 6,where z is a digit what is the value of z
This time there is a solution, assuming the expression is supposed to be 41z^2. 41 is not a multiple of 6; but 6 is. So for this one the answer is z=6.
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
(1) if 41x is a multiple of 11, where x is a digit what is the value of x
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The divisibility by 11 rule 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 Divisibility by 11 rule 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.
Solved.
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(2) if 3y5 is a multiple of 11 , where y is a digit what is the value of y.
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.
Solved.
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(3) if 41z2 is a multiple of 6, where z is a digit what is the value of z.
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.
Solved.
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On divisibility rules for 2, 3 and 6 read in my lessons
- Divisibility by 2 rule
- Divisibility by 3 rule
- Divisibility by 6 rule
in this site.
For other problems closely related to your in this post, see the lessons
- Restore the omitted digit in a number in a way that the number is divisible by 9 and
- Restore the omitted digit in a number in a way that the number is divisible by 11.
in this site.
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