SOLUTION: the number 1358*16, where * represents a digit, is exactly divisible by 11 then, then what digit is represented by *?

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Question 992184: the number 1358*16, where * represents a digit, is exactly
divisible by 11 then, then what digit is represented by *?
Found 3 solutions by solver91311, Edwin McCravy, ikleyn:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Where did you obtain the utterly ridiculous notion that 11 is a factor of the number 1358*16? If your math teacher told you that, get a new math teacher.

John

My calculator said it, I believe it, that settles it

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

Pay no attention to Mr. Grouchy Grumpypants :)
above. I knew exactly what you meant.  I 
rewrote it so he could understand.

1358*16

Let the digit represented by * be N

Then 1358*16 = 1358000 + 100N + 16

When we divide that by 11 we must get an 
integer:

%281358000%2B100N%2B16%29%2F11 must equal an 
integer, say integer I

%281358000%2B100N%2B16%29%2F11%22%22=%22%22I

1358000%2F11%2B100N%2F11%2B16%2F11%22%22=%22%22I

Since 1358000%2F11=123454.5455, the 
largest multiple of 11 that does not exceed 
1358000 is 123454×11 or 1357994, and since
1358000-1357994 = 6 we write 1358000 as 
1357994+6.

Since the largest multiple of 11 that does not 
exceed 100 is 99, we write 100N as 99N+N.

Since the largest multiple of 11 that does 
not exceed 16 is 11, we write 16 as 11+5

%281357994%2B6%29%2F11%2B%2899N%2BN%29%2F11%2B%2811%2B5%29%2F11%22%22=%22%22I

1357994%2F11%2B6%2F11%2B99N%2F11%2BN%2F11%2B11%2F11%2B5%2F11%22%22=%22%22I

123454%2B6%2F11%2B9N%2BN%2F11%2B1%2B5%2F11%22%22=%22%22I

Leave the fractions on the left, move the 
whole terms right

6%2F11%2BN%2F11%2B5%2F11%22%22=%22%22I-123454-9N-1%2B5

%286%2BN%2B5%29%2F11%22%22=%22%22I-123450-9N

%2811%2BN%29%2F11%22%22=%22%22I-123450-9N

The right side is an integer so the left side 
must be too.

The only digit that will cause the left side 
to be an integer is 

N = 0

So the asterisk * must represent 0.

Checking:  When we divide 1358016 by 11 we 
get 123456

Edwin

Answer by ikleyn(52806) About Me  (Show Source):
You can put this solution on YOUR website!
.
Hello,
there is much shorter solution.

The  "divisibility by 11 rule"  says:  the number is divisible by  11  if and only if the alternate sum of its digits is divisible by  11.
See the lesson  Divisibility by 11 rule  in this site.

In our case the alternate sum of digits is

1 - 3 + 5 - 8 + x - 1 + 6 = -5 + x + 5 = x,

where  x  stands instead of asterisk.

So,  x  should be zero to be divisible by  11, and,  hence, our number is 1358016.