Difference between revisions of "2019 Mock AMC 10B Problems/Problem 19"
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<math>3^{1008} + 1 \equiv 3^{504} + 1 \equiv 3^{252} + 1 \equiv 3^{126} + 1 \equiv 3^{63} - 1 \equiv 2</math> <math>\text{mod}</math> <math>4</math>, and <math>3^{63} + 1 \equiv 4</math> <math>\text{mod}</math> <math>8</math>. | <math>3^{1008} + 1 \equiv 3^{504} + 1 \equiv 3^{252} + 1 \equiv 3^{126} + 1 \equiv 3^{63} - 1 \equiv 2</math> <math>\text{mod}</math> <math>4</math>, and <math>3^{63} + 1 \equiv 4</math> <math>\text{mod}</math> <math>8</math>. | ||
− | Therefore, the smallest powers of <math>2</math> that divide each of these numbers are <math>2, 2, 2, 2, 2</math>, and <math>4</math> | + | Therefore, the smallest powers of <math>2</math> that divide each of these numbers are <math>2, 2, 2, 2, 2</math>, and <math>4</math>. The smallest power of <math>2</math> that divides <math>3^{2016} - 1</math> is thus <math>2^5 \cdot 4 = \boxed{\text{(E)} 256}</math>. |
<baker77> | <baker77> |
Revision as of 21:55, 2 November 2019
Problem
What is the largest power of that divides
?
Solution
.
By simple mod checking, we find that
, and
.
Therefore, the smallest powers of that divide each of these numbers are
, and
. The smallest power of
that divides
is thus
.
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