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#1 2006-10-09 08:44:20
- cat908111
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- Registered: 2006-10-09
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digits
The numbers A and B each have three digits. Fred was asked to calculate A x B. Instead he put A on the left of B to form a six digit number D. His answer was 3 times the correct answer. Find A and B. Justify why there is only one solution.
Can anyone help???
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#2 2006-10-09 09:20:32
- mathsyperson
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Re: digits
This is the same as saying that 1000a + b = 3ab.
Rearranging gives 1000a - 3ab = -b
a(1000-3b) = -b
a = b/(3b-1000)
a is a 3-digit whole number, which means that b/(3b-1000) also has to be a 3-digit whole number.
It is obvious that b cannot be 333 or less, because then the term would be negative. Also, it cannot be too much more than 333, because then the term would be too small to be a 3-digit number. If b=334, then a = 334/2 = 167. If b = 335, then a = 335/5 = 67, which is too small. And so all higher values will return a-values that are too small as well.
Therefore, the two numbers must be 167 and 334.
Check: 167*334 = 55778.
55779*3 = 167334.
There are a few more solutions where A is a 2-digit number, if you're interested.
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#3 2006-10-09 13:54:00
- John E. Franklin
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Re: digits
Excellent work except for one comment that may be a typo on line 5 or 7, depending if you count blank rows.
"...number, which means that b/(1000-3b) also..."
Why did you negatize the denominator?? by mistake??
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#4 2006-10-09 20:27:00
- mahmoudaljamel
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Re: digits
1000a + b = 3ab
1000a - 3ab = -b
a( 1000 - 3b) = -b
a ( 3b - 1000) = b
a = b / ( 3b - 1000)
If you always do what you always did, you'll always get what you always got
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#5 2006-10-09 22:33:35
- mathsyperson
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Re: digits
Ah yes, thanks John. I worked the question through with the negative denominator like that and got an answer of 333 and 333, which was wrong. I then realised my mistake and corrected it, but forgot to correct the subsequent one.
I think I'll just do a silent edit and pretend that never happened. 
Why did the vector cross the road?
It wanted to be normal.
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#6 2006-10-13 20:49:03
- cat908111
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Re: digits
why use 1000 when using the algebraic method??
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#7 2006-10-13 22:02:47
- luca-deltodesco
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- Registered: 2006-05-05
- Posts: 1,470
Re: digits
you use 1000, because multiplying a number by 1000 will shift it 3 digits to the left, so for example
a = 312, b = 124
1000a = 312000, so 1000a + b = 312124, which is what was needed for 'he put it to the left of the other number'
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