Difference between revisions of "2019 AMC 10B Problems/Problem 14"
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==Solution== | ==Solution== | ||
− | We can figure out H = 0 by noticing that 19! will end with 3 zeroes, as there are three 5's in its prime factorization. Next we use the fact that 19! is a multiple of both 11 and 9. | + | We can figure out <math>H = 0</math> by noticing that <math>19!</math> will end with <math>3</math> zeroes, as there are three <math>5</math>'s in its prime factorization. Next we use the fact that <math>19!</math> is a multiple of both <math>11</math> and <math>9</math>. Since their divisibility rules gives us that <math>T + M</math> is congruent to <math>3</math> mod <math>9</math> and that <math>T - M</math> is congruent to <math>7</math> mod <math>11</math>. By inspection, we see that <math>T = 4, M = 8</math> is a valid solution. Therefore the answer is <math>4 + 8 + 0 = 12</math>, which is (C). |
− | - AZAZ12345 | + | - AZAZ12345 |
+ | |||
+ | Edited (LaTeXed) by greersc | ||
==See Also== | ==See Also== |
Revision as of 17:11, 14 February 2019
Problem
The base-ten representation for is , where , , and denote digits that are not given. What is ?
Solution
We can figure out by noticing that will end with zeroes, as there are three 's in its prime factorization. Next we use the fact that is a multiple of both and . Since their divisibility rules gives us that is congruent to mod and that is congruent to mod . By inspection, we see that is a valid solution. Therefore the answer is , which is (C).
- AZAZ12345
Edited (LaTeXed) by greersc
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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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