The problem states that the product of the digits of a two-digit number exceeds their sum by 39, and reversing the digits increases the number by 27. By representing the digits with variables x and y, two equations are formed and solved using algebra to obtain x=6 and y=9, so the number is 69.

1. Let represent the tens digit and represent the units digit. Then the information given can
be translated mathematically as and .
Subtracting from both sides of the second equation gives us . Dividing
by yields . Substituting for in the first equation gives us
. Expanding and grouping like terms yields .
Factoring this equation, we get . Since must be positive, we get that
. Since , . So our number must be 69.
Subscribe to comment feed
Featured Solution
Submitted by Calvin L.
x y
xy = x + y + 39 10y + x = 10x + y + 27
x + y 9y = 9x + 27
9 y = x + 3 x + 3 y
x(x + 3) = x + (x + 3) + 39 + x − 42 = 0x2
(x + 7)(x − 6) = 0 x
x = 6 y = x + 3 y = 9
Gabriel B. 20, Brazil May 28 at 04:12am
Jembatan G, see: 5+8 = (5 x 8) -39 (FALSE)
Calvin L. , USA May 17 at 05:47am
@Jembatan Please read the question. It says "product exceeds the sum of digits by 39",
meaning that .
You wrote down , and I'm not sure how you reached your interpretation. This
would correspond to "sum of digits is more than the sum of digits of the number 39".
x × y = x + y + 39
x + y > 3 + 9
Details and assumptions
The number is a 2-digit number, not a 3-digit number.
KEY TECHNIQUES
Number Base Representation, Quadratic Formula (Roots of Quadratic Equation)
ANSWER
69
The product of the digits of a positive two-digit number exceeds the sum of the digits by . If the order of
the digits is reversed, the number is increased by . Find the number.
39
27
12 = 012
Let represent the tens digit and represent the units digit. Then the information given can be translated mathematically as
and . Subtracting from both sides of the second equation gives us . Dividing by yields
. Substituting for in the first equation gives us . Expanding and grouping like terms yields
. Factoring this equation, we get . Since must be positive, we get that . Since , .
So our number must be 69.
x y xy = x + y + 39
10y + x = 10x + y + 27 x + y 9y = 9x + 27 9
y = x + 3 x + 3 y x(x + 3) = x + (x + 3) + 39
+ x − 42 = 0x2
(x + 7)(x − 6) = 0 x x = 6 y = x + 3 y = 9
The Digits
55 points
Remember to put math in brackets like ( 2 + 2 ) or ( 2^{34} )
Post CommentPreview
Write a comment or ask a question...
××
Formatting guide
Brilliant
Page 1 of 2The Digits | Brilliant
6/21/2013https://brilliant.org/assessment/s/algebra-and-number-theory/1729233/

2. Get more free challenging math and physics problems at briliant.org
Jembatan G. 13, Indonesia May 17 at 05:40am
The answer can be 58 too. Count it. 5+8>3+9. 5 & 8 reverse. 85. 85 - 58 = 27
Affan W. 18, Pakistan May 13 at 04:43pm
you people ROCK!!!
Andrias Y. 20, Indonesia May 13 at 01:24pm
thanks for the solution :))
Page 2 of 2The Digits | Brilliant
6/21/2013https://brilliant.org/assessment/s/algebra-and-number-theory/1729233/