K is a factor of 30! + 40!. If K is a prime number less than 40, the

Question

K is a factor of 30! + 40!. If K is a prime number less than 40, the highest possible value of K is?

A. 19
B. 23
C. 29
D. 31
E. 37

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Dereno · Answer

K is a Factor of 30! + 40!.

k is a prime number less than 40.

Highest K = ?

30! + 40! = 30! * ( 1+ 31*32*33*34*35*36*37*38*39*40)

when this expression is divisible by K , the remainder should be zero.

30 ! = 1*2*3*4*5*6*7.....27*28*29*30

The highest prime number in 30! is 29.

This expression : ( 1+ 31*32*33*34*35*36*37*38*39*40) contains (1+31p) OR (1+37p). When divided by 31 and 37, we get a remainder of 1. Hence not divisible.

29 is a prime number greater than 23 and 19. Hence,  Option C (29) .

vasu1104 · Answer

30! (1 + 40*39*38*37*36*35*34*33*32*31)

now K is prime number less than 40.
and k is factor of above mentioned sequence. so from there we can say k is factor of 30!.
and the largest prime number in that sequence will be 29.
option C

sujoykrdatta · Answer

30! + 40! = 30! x (1 + 40 ×39×38×37...31)

It is clear that 30! has 29 as a factor.  
 Also, 40×39×38×37...31 is a multiple of both 31 and 37.  
 Thus, 40×39×38×37...31 + 1 is NEITHER divisible 31 nor 37 
 Thus, the highest prime factor from among the options is 29.

Answer C

Note : A point to be noted is that the value (1 + 40×39×38×37...31) will have prime factors as well ( which is difficult to determine manually here and is beyond the scope of GMAT ), but it will definitely have prime factors greater than 37. Thus, the highest value of K need not be 29. The highest value of K  smaller than 40  is definitely 29.

ExpertsGlobal5 · Answer

Video explanation:

https://www.youtube.com/watch?v=EYSZPa2Wxv8

K is a factor of 30! + 40!. If K is a prime number less than 40, the

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K is a factor of 30! + 40!. If K is a prime number less than 40, the

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