Pretty Properties
Let be the set of all two-digit integers for which the product of their digits is exactly twice as much as the sum of their digits.
Let be the set of all two-digit integers which become 75% greater when the digits are flipped.
Let be the number of two-digit integers in the union .
Let be the only two-digit integer that is a member of the intersection .
How much is ?
The answer is 42.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Set A :
If the number is x y then x y = 2 ( x + y ) ∴ x y − 2 x − 2 y = 0 ∴ ( x − 2 ) ( y − 2 ) = 4 . Since 4 = 1 ⋅ 4 = 2 ⋅ 2 = 4 ⋅ 1 , we have solutions x y = 3 6 , 4 4 , 6 3 .
Set B :
If the number is x y then y x = 4 7 x y . This means 7 ( 1 0 x + y ) = 4 ( 1 0 y + x ) ∴ 6 6 x = 3 3 y ∴ 2 x = y . With 1 ≤ x , y ≤ 9 , this gives us x y = 1 2 , 2 4 , 3 6 , 4 8 .
The union is A ∪ B = { 3 6 , 4 4 , 6 3 } ∪ { 1 2 , 2 4 , 3 6 , 4 8 } = { 1 2 , 2 4 , 3 6 , 4 4 , 4 8 , 6 3 } ; this shows n = 6 .
The intersection A ∩ B = { 3 6 } , so that x = 3 6 .
The answer is therefore n + x = 6 + 3 6 = 4 2 .