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31 Mar 2022, 03:15
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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The value of \(a^*\) is the product of all positive, odd integers less then \(a\) but greater than 0. If \(e = 16\) and \(o = 14\) what is the largest prime factor of \(e^* + 2o^*\)?
(A) 23
(B) 19
(C) 17
(D) 13
(E) 11
(A) 23
(B) 19
(C) 17
(D) 13
(E) 11
Bunuel
The value of \(a^*\) is the product of all positive, odd integers less then \(a\) but greater than 0. If \(e = 16\) and \(o = 14\) what is the largest prime factor of \(e^* + 2o^*\)?
(A) 23
(B) 19
(C) 17
(D) 13
(E) 11
(A) 23
(B) 19
(C) 17
(D) 13
(E) 11
\( a^* =1 *3*5*7*9*11*13*15\)... (i)
\((2o)^* = (2*14 )^* = 28^*\)
Hence \( (2o)^*=28^* = (e^*) * 17 *19 *21*23*25*27\)... (ii)
Required highest prime from : \((a^*) + (2o)^*\)
\(=(a^*) + (a^*) * 17 *19 *21*23*25*27 \)... Substituting the value of \((2o)^* \)from (ii)
\(=(a^*)*(1 + 17 *19 *21*23*25*27)\)... Taking common factor \((a^*) \) out.
\(=(1*3*5*7*9*11*13*15)*(1 + 17 *19 *21*23*25*27 ) \)...Substituing the value of \((a^*)\) from (i)
Notice the expression: \(( 1 + 17 *19 *21*23*25*27 ) = \) (odd + odd ) an even integer greater than \(2 \)
Hence this expr : \(( 1 + 17 *19 *21*23*25*27 )\)
Edit : We can still factor out a prime from the exp. Turns out I misinterpreted the question
Please refer to ensuing post by Ian sir for the correct soluton. Thanks to Ian sir.
Highest prime\( = 13\)
Ans D
Hope it's clear.
07 Apr 2022, 10:29
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From the notation, 16* is equal to the product of the odd numbers from 1 through 15, and 14* is equal to the product of the odd numbers from 1 through 13. Notice that 16* = (15)(13)(11)...(3)(1) = (15)(14*). So 16* + 2(14*) = 15(14*) + 2(14*) = 17(14*), and since 14* isn't divisible by any prime greater than 13, the answer is 17.
stne -- I think in your solution, you have solved as if the question asked about \(e^* + (2o)^*\), but here the star only applies to what its immediately next to, just as a factorial symbol would; 2n! is equal to 2 times n!, and not to (2n) factorial.
stne -- I think in your solution, you have solved as if the question asked about \(e^* + (2o)^*\), but here the star only applies to what its immediately next to, just as a factorial symbol would; 2n! is equal to 2 times n!, and not to (2n) factorial.
