Difference between revisions of "2021 Fall AMC 10B Problems/Problem 6"
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==Solution 2== | ==Solution 2== | ||
− | Recall that <math>6^k</math> can be written as <math>2^k \cdot 3^k</math>. Since we want the integer to have <math>2021</math> divisors, we must have it in the form <math>p_1^42 \cdot p_2^46</math>, where <math>p_1</math> and <math>p_2</math> are prime numbers. Therefore, we want <math>p_1</math> to be <math>3</math> and <math>p_2</math> to be <math>2</math>. To make up the remaining <math>2^4</math>, we multiply <math>2^42 \cdot 3^42</math> by <math>m</math>, which is <math>2^4</math> which is 16. Therefore, we have <math>42 + 16 = \boxed {(B) 58}</math> | + | Recall that <math>6^k</math> can be written as <math>2^k \cdot 3^k</math>. Since we want the integer to have <math>2021</math> divisors, we must have it in the form <math>p_1^42 \cdot p_2^46</math>, where <math>p_1</math> and <math>p_2</math> are prime numbers. Therefore, we want <math>p_1</math> to be <math>3</math> and <math>p_2</math> to be <math>2</math>. To make up the remaining <math>2^4</math>, we multiply <math>2^{42} \cdot 3^{42}</math> by <math>m</math>, which is <math>2^4</math> which is <math>16</math>. Therefore, we have <math>42 + 16 = \boxed {(B) 58}</math> |
~Arcticturn | ~Arcticturn |
Revision as of 10:53, 23 November 2021
Contents
[hide]Problem
The least positive integer with exactly distinct positive divisors can be written in the form , where and are integers and is not a divisor of . What is
Solution
Let this positive integer be written as . The number of factors of this number is therefore , and this must equal 2021. The prime factorization of 2021 is , so and . To minimize this integer, we set and . Then this integer is . Now and so
~KingRavi
Solution 2
Recall that can be written as . Since we want the integer to have divisors, we must have it in the form , where and are prime numbers. Therefore, we want to be and to be . To make up the remaining , we multiply by , which is which is . Therefore, we have
~Arcticturn
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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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