If x is an integer, is 2^x a factor of 12!?

Question

If x is an integer, is 2^x a factor of 12! ?

(1) x is the sum of two distinct single-digit prime numbers.
(2) 0 < x < 11

LevanKhukhunashvili · Answer

12! to be divisible by  2^x , x must not be more than 10

1 stm--> 2+3<10; 5+7>10  not suff 
2 stm --> favorable range.  suff

IMO
Ans:  B

Archit3110 · Answer

factor of 12!/2^x
6+3+1; 10 factors possible
#1
x is the sum of two distinct single-digit prime numbers
3+7 = 10; or say 7+5;12 insufficient
#2
0 < x < 11[
x=10 ; 2^10 sufficient
IMO B

ak31 · Answer

Regarding your first equation, why can't x be 7+5 =12?

queensbridge · Answer

Because the prime factorisation of 12! only includes 10 2's and therefore X cannot contain more than this. The answer is  B .

Aprajita760 · Answer

Hello,
Can you please explain how are we getting 10 factors for 12!?

aliakberza · Answer

Aprajita760 12! = 12*11*10*9*8*7*6*5*4*3*2*1 For 2^x to be a factor of 12!, 12! needs to be fully divisible by 2^x (i.e. no remainder). If we break down the 12! sequence into prime factors (e.g. 12=2^2*3^1 ) we would get: (2^2*3^1)(11^1)(2^1*5^1)(3^2)(2^3)(7^1)(2^1*3^1)(5^1)(2^2)(3^1)(2^1)(2^0) , which simplifies to: 2^10* 3^5*5^2*7^1*11^1 . Now, we can see that 2^10 is the largest power of 2 that is a factor of 12!. If we know whether x is more or less than 10 we will know whether 2^x is a factor of 12!. St-1: x is the sum of two distinct prime factors: if the prime factors are 2 and 3 then x=5 and 2^5 is a factor of 12!, but if the prime factors are 5 and 7 then x=12 and 2^12 is NOT a factor of 12!. Insufficient St-2: 0 < x < 11: As x is less than 11 and an integer (question stem) we know that the max value of x can be 10 and based on the above we know that 2^10 is a factor of 12!. Sufficient Answer is therefore B Hope this helps

LevanKhukhunashvili · Answer

aliakberza

How to find maximum power of 2 in 12!

We have to divide 12 by every  2^x , where x is a positive integer and  2^x<12 , ignore the remainder and keep the quotient

12/ 2^1 =6
12/ 2^2 =3
12/ 2^3 =1

6+3+1=10

So the maximum value of x can be 10

Hope, it's clear now

CareerGeek · Answer

12! = 2*4*6*8*10*12*k, where k is an odd integer —> 12! = 2*2^2*2*2^3*2*2^2*m where m is any odd integer —> 12! = 2^10*m So maximum value of x for which 2^x can be a factor of 12! is “10” (1) x is the sum of two distinct single-digit prime numbers. Maximum value of x possible = 5 + 7 = 12 —> x > 10 Insufficient (2) 0 < x < 11 —> Maximum value of x possible = 10 Sufficient IMO Option B Pls Hit kudos if you like the solution Posted from my mobile device

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If x is an integer, is 2^x a factor of 12!?

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If x is an integer, is 2^x a factor of 12!?

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