# Difference between revisions of "2004 Pan African MO Problems/Problem 2"

(Created page with "<math>\sqrt{4-2\sqrt{3}} = a\sqrt{3}-b</math>. Through guess and check with small numbers, <math>a = 1</math> and <math>b = 1</math>. So <math>\sqrt{4-2\sqrt{3}} = \sqrt{3}-...") |
Rockmanex3 (talk | contribs) m (Reformatting) |
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+ | ==Problem== | ||

+ | |||

+ | Is <math>4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}}</math> an integer? | ||

+ | |||

+ | ==Solution== | ||

+ | |||

<math>\sqrt{4-2\sqrt{3}} = a\sqrt{3}-b</math>. | <math>\sqrt{4-2\sqrt{3}} = a\sqrt{3}-b</math>. | ||

Through guess and check with small numbers, <math>a = 1</math> and <math>b = 1</math>. | Through guess and check with small numbers, <math>a = 1</math> and <math>b = 1</math>. | ||

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So <math>\sqrt{97-56\sqrt{3}} = 7-4\sqrt{3}</math>. | So <math>\sqrt{97-56\sqrt{3}} = 7-4\sqrt{3}</math>. | ||

− | Value of <math>4\sqrt{4-2\sqrt{3}} + \sqrt{97-56\sqrt{3}} = (4\sqrt{3}-4) + (7-4\sqrt{3}) = 3</math> | + | Value of <math>4\sqrt{4-2\sqrt{3}} + \sqrt{97-56\sqrt{3}} = (4\sqrt{3}-4) + (7-4\sqrt{3}) = 3</math>. |

+ | |||

+ | ==See Also== | ||

+ | {{Pan African MO box|year=2004|num-b=1|num-a=3}} | ||

+ | |||

+ | [[Category:Introductory Number Theory Problems]] |