Difference between revisions of "2005 AMC 10B Problems/Problem 24"
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== Solution == | == Solution == | ||
− | Let <math>x = 10a+b, y = 10b+a</math>, [[without loss of generality]] with <math>a>b</math>. Then <math>x^2 - y^2 = (x-y)(x+y) = (9a - 9b)(11a + 11b) = 99(a-b)(a+b) = m^2</math>. It follows that <math>11|(a-b)(a+b)</math>, but <math>a-b < 10</math> so <math>11|a+b \Longrightarrow a+b=11</math>. Then we have <math>33^2(a-b) = m^2</math>. Thus <math>a-b</math> is a perfect square. Also, | + | Let <math>x = 10a+b, y = 10b+a</math>, [[without loss of generality]] with <math>a>b</math>. Then <math>x^2 - y^2 = (x-y)(x+y) = (9a - 9b)(11a + 11b) = 99(a-b)(a+b) = m^2</math>. It follows that <math>11|(a-b)(a+b)</math>, but <math>a-b < 10</math> so <math>11|a+b \Longrightarrow a+b=11</math>. Then we have <math>33^2(a-b) = m^2</math>. Thus <math>a-b</math> is a perfect square. Also, because <math>a-b</math> and <math>a+b</math> have the same parity, <math>a-b</math> is a one-digit odd perfect square, namely <math>1</math> or <math>9</math>. The latter case gives <math>(a,b) = (10,1)</math>, which does not work. The former case gives <math>(a,b) = (6,5)</math>, which works, and we have <math>x+y+m = 65 + 56 + 33 = 154\ \mathbf{(E)}</math>. |
== Solution 2 == | == Solution 2 == |
Revision as of 19:32, 20 April 2014
Contents
Problem
Let and
be two-digit integers such that
is obtained by reversing the digits
of
. The integers
and
satisfy
for some positive integer
.
What is
?
Solution
Let , without loss of generality with
. Then
. It follows that
, but
so
. Then we have
. Thus
is a perfect square. Also, because
and
have the same parity,
is a one-digit odd perfect square, namely
or
. The latter case gives
, which does not work. The former case gives
, which works, and we have
.
Solution 2
The first steps are the same as above. Let , where we know that a and b are digits (whole numbers less than 10). Like above, we end up getting
. This is where the solution diverges.
We know that the left side of the equation is a perfect square because m is an integer. If we factor 99 into its prime factors, we get . In order to get a perfect square on the left side,
must make both prime exponents even. Because the a and b are digits, a simple guess would be that
(the bigger number) equals 11 while
is a factor of nine (1 or 9). The correct guesses are
causing
and
. The sum of the numbers is
See Also
2005 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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