Difference between revisions of "1974 AHSME Problems/Problem 26"
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==Solution== | ==Solution== | ||
| − | The prime factorization of <math> 30 </math> is <math> 2\cdot3\cdot5 </math>, so the prime factorization of <math> 30^4 </math> is <math> 2^4\cdot3^4\cdot5^4 </math>. Therefore, the number of positive divisors of <math> 30^4 </math> is <math> (4+1)(4+1)(4+1)=125 </math>. However, we have to subtract <math> 2 </math> to account for <math> 1 </math> and <math> 30^4 </math>, so our final answer is | + | The prime factorization of <math> 30 </math> is <math> 2\cdot3\cdot5 </math>, so the prime factorization of <math> 30^4 </math> is <math> 2^4\cdot3^4\cdot5^4 </math>. Therefore, the number of positive divisors of <math> 30^4 </math> is <math> (4+1)(4+1)(4+1)=125 </math>. However, we have to subtract <math> 2 </math> to account for <math> 1 </math> and <math> 30^4 </math>, so our final answer is $ 125-2=123, \boxed{\text{C}} |
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| + | soln by RNVAA | ||
==See Also== | ==See Also== | ||
Latest revision as of 06:55, 31 August 2024
Problem
The number of distinct positive integral divisors of 
excluding 
and is
Solution
The prime factorization of 
is 
, so the prime factorization of 
is 
. Therefore, the number of positive divisors of is 
. However, we have to subtract 
to account for and , so our final answer is $ 125-2=123, \boxed{\text{C}}
soln by RNVAA
See Also
| 1974 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 25 |
Followed by Problem 27 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
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