Lesson Product of Digits Problem

Source code of 'Product of Digits Problem'

This Lesson (Product of Digits Problem) was created by by oberobic(2304)

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About oberobic: MBA/Ph.D. University Administrator

Problem: A two-digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their positions. Find the numbers.
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Solution: With digit problems like this, you have to look at a number in terms of its appearance as well as the place values of its digits. Consider the number 'xy'. The number simply looks like 'xy', but the value of 'xy' is 10*x + 1*y.
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Note well that 'xy' does not mean x*y, but rather the visual representation of 'x' sitting next to 'y'.
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For this problem, we are told the product of its digits is 18:
x*y = 18
so
x = 18/y
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Recall the value of the 'xy' is: 10x+y
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We also are told that when 63 is subtracted from the number, the digits are reversed.
This means xy -63 = yx
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Remember that 'xy' and 'yx' are not multiplications. The numbers are just sitting beside one another.
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So, when 63 is subtracted from the value of the number, the digits are reversed.
10x + y -63 = 10y + x
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Substitute for 'x' to convert the equation to have only one unknown.
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10*18/y +y -63 = 10y +18/y
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180/y + y -63 = 10y + 18/y
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Multiply both sides by y to eliminate the fractions.
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180 + y^2 -63y = 10y^2 + 18
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Collect terms.
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10y^ + 18 = y^2 -63y + 180
9y^2 +63y -162 = 0
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Divide both sides by 9. Or course, 0/9 is just 0.
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y^2 +7y - 18 = 0
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Factor the quadratic equation if possible.
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(y+9)(y-2) = 0
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y = -9 or 2
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Substitute y=2 to find x.
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x*2 = 18
x = 9
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The original number appears to be: 92.
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Always check your work.
The product of 9*2 = 18, which is how the solution was found.
To check these values, you have to use the other equation.
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92 - 63 = 29
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Great! The digits are interchanged.
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Answer: The original number is 92. The other number is 29.
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Postscript: But what about the y=-9? Why didn't we use it, too?
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Well, -9 takes up two spaces when you type it, so it is not a single digit. But the algebra doubtless works, or the solution would be wrong. 
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x*-9 = 18
x = -2
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-29 -63 = -92
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So, the algebra works just fine. But when we say a "two-digit" number, we would not want it to take 3 or 4 spaces when typed out. We would need a means to show a negative number with a different symbol that takes only one space for this solution to make common sense.