SOLUTION: If the positive integers a and b satisfy √a − √b = √11, what is the maximum value of a/b?

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Question 1162750: If the positive integers a and b satisfy √a − √b = √11, what is the maximum value of a/b?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13208) About Me  (Show Source):
You can put this solution on YOUR website!


The following original response left out important parts of the solution -- though the final answer to the question was correct. See further down for the corrected/expanded response.

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If a and b are integers and sqrt%28a%29-sqrt%28b%29+=+sqrt%2811%29, then for some positive integer value of n,

a=%28n%2B1%29%5E2 and b+=+n%5E2

Then a%2Fb+=+%28n%2B1%29%5E2%2Fn%5E2+=+%28%28n%2B1%29%2Fn%29%5E2

For positive integer values of n, the maximum value of a/b is clearly when n=1 and a/b = (2/1)^2 = 4.

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Corrected response....

If a and b are integers and sqrt%28a%29-sqrt%28b%29+=+sqrt%2811%29, then for some positive integer value of n,

a=%2811%28n%2B1%29%5E2%29 and b+=+11%28n%5E2%29

With those values of a and b,
sqrt%28a%29+=+%28n%2B1%29%2Asqrt%2811%29
sqrt%28b%29+=+%28n%29%2Asqrt%2811%29
sqrt%28a%29-sqrt%28b%29+=+%28n%2B1%29%2Asqrt%2811%29-%28n%29%2Asqrt%2811%29+=+sqrt%2811%29

Then a%2Fb+=+%28n%2B1%29%5E2%2Fn%5E2+=+%28%28n%2B1%29%2Fn%29%5E2+=+%281%2B1%2Fn%29%5E2

For positive integer values of n, the maximum value of a/b is clearly when n=1 and a/b = (2/1)^2 = 4.

CHECK....

n=1: a = 11(2^2) = 44; b = 11(1^2) = 11; sqrt%28a%29-sqrt%28b%29+=+2%2Asqrt%2811%29-1%2Asqrt%2811%29+=+sqrt%2811%29; a/b = 44/11 = 4

n=2: a = 11(3^2) = 99; b = 11(2^2) = 44; sqrt%28a%29-sqrt%28b%29+=+3%2Asqrt%2811%29-2%2Asqrt%2811%29+=+sqrt%2811%29; a/b = 99/44 = 9/4

n=3: a = 11(4^2) = 176; b = 11(3^2) = 99; sqrt%28a%29-sqrt%28b%29+=+4%2Asqrt%2811%29-3%2Asqrt%2811%29+=+sqrt%2811%29; a/b = 176/99 = 16/9


Clearly for larger values of n the ratio a/b will continue to get smaller....

Answer by ikleyn(52875) About Me  (Show Source):
You can put this solution on YOUR website!
.

To tutor @greenestamps.


Your assumption that the numbers  "a"  and  "b"  are perfect squares,  contradicts  to the given equation  (!)