Difference between revisions of "2007 iTest Problems/Problem 20"
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Rockmanex3 (talk | contribs) (Solution to Problem 20) |
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− | == Problem == | + | ==Problem== |
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Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math> | Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math> | ||
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\text{(T) } 2007\qquad</math> | \text{(T) } 2007\qquad</math> | ||
− | == Solution == | + | ==Solution== |
+ | |||
+ | The expression can be factored by repeatedly using the difference of squares. | ||
+ | <cmath>(2007^{512} + 1)(2007^{512} - 1)</cmath> | ||
+ | <cmath>(2007^{512} + 1)(2007^{256} + 1)(2007^{256} - 1)</cmath> | ||
+ | <cmath>(2007^{512} + 1)(2007^{256} + 1) \cdots (2007^1 + 1)(2007^1 - 1)</cmath> | ||
+ | Notice that <math>2007 \equiv 3 \pmod{4}</math>, so <math>2007^2 \equiv 1 \pmod{4}</math>. Thus, in the expression <math>2007^a + 1</math>, if <math>a</math> is even, then the expression is congruent to <math>2</math> [[modulo]] <math>4</math>. | ||
+ | |||
+ | <br> | ||
+ | The remaining numbers to consider are <math>2008</math> and <math>2006</math>. Factoring <math>2008</math> yields <math>8 \cdot 251</math>, and factoring <math>2006</math> yields <math>2 \cdot 1003</math>. | ||
+ | |||
+ | <br> | ||
+ | That means <math>2007^{1004} - 1</math> has <math>9+3+1 = 13</math> as the exponent of <math>2</math>, so the largest <math>n</math> that makes <math>2^n</math> a factor of <math>2007^{1004} - 1</math> is <math>\boxed{\text{(M) } 13}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{iTest box|year=2007|num-b=19|num-a=21}} | ||
+ | |||
+ | [[Category:Intermediate Number Theory Problems]] |
Latest revision as of 21:23, 28 July 2018
Problem
Find the largest integer such that is divisible by
Solution
The expression can be factored by repeatedly using the difference of squares. Notice that , so . Thus, in the expression , if is even, then the expression is congruent to modulo .
The remaining numbers to consider are and . Factoring yields , and factoring yields .
That means has as the exponent of , so the largest that makes a factor of is .
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 19 |
Followed by: Problem 21 | |
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