x is a positive integer such as 4,095

Question

x is a positive integer such as 4,095<x<4,105. What is x?
(1) When x is divided by 2 or by 3, the remainder is 1.
(2) x is a prime number.

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SaidNassar1991 · Answer

(1) The remainder of x when divided by 2 =1 means x is odd. So x is 4,097 or 4,099 or 4,101 or 4,103 . When divided by 3 remainder is 1: the multiples of 3 between 4,095 and 4.105 are 4,095 and 4,098 and 4,101 and 4,104 so x could be 4,096 or 4,099 or 4,102. The intersection of the two conditions is 4,099. So (1) is sufficient

(2) The numbers between 4,095 and 5,095 that are not multiples of 2 and 3 and then could be a prime number are 4,097 and 4,099 and 4,103.
To find if they are prime or no we should try dividing them by the prime numbers other than 2 and 3 and 5 {7,11,13,17...}

To check if 4,097 and 4,099 and 4,103 are divisible by 7, we should find the closest multiple of 7.. 7*500=3500; 3500+7*50=3850; 3,850+7*35=4,095 and 4,102 so they r not divisible by 7.

To check if 4,097 and 4,099 and 4,103 are divisible by 11, we should find the closest multiple of 11

11*400=4,400; 4,400-11*30= 4,070; 4,070+11*3= 4,103 so 4,103 is not a prime.
We still have 4,097 and 4,099.

To check if 4,097 and 4,099 are divisible by 13, we should find the closest multiple of 13.
13*300=3,900; 3,900+13*15=4,095. So 4,097 and 4,099 are not divisible by 13.

To check if 4,097 and 4,099 are divisible by 17, we should find the closest multiple of 17.
17*200=3,400; 3,400+17*40=4,080,4,080+17=4,097 so 4,097 is not Prime. “x is prime” then x=4,099. So it’s sufficient.

Answer is D.

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globaldesi · Answer

the number of odd numbers in between are

4097
4099
4101
4103
thus

only 4099 gives remainder 1 when divided by 2 and 3
thus x = 4099

for x to be prime 
this was little calculation centric but then 4099 is not divisible by any number upto 21 thus it has tp eb D
i am not sure of any shorter way

SaidNassar1991 · Answer

globaldesi  it’s not sufficient to just arrive to 21, in fact 21*21 is 441 and not bigger than 4,000

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tapovrat · Answer

I chose D but took me 3.5 minutes in all.
Any idea how to solve for the second statement most efficiently?

Thanks in advance!

SaidNassar1991 · Answer

tapovrat  did u read my reply? This method takes less than 2 mins all in all

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tapovrat · Answer

Hi Said,

Yes, infact I followed the exact same approach as yours. But I started checking prime/not from 4099 and hence wasted too much time there. 
Any particular reason why you picked the ascending order to check prime divisibility?

SaidNassar1991 · Answer

Hi  tapovrat , it’s so hard to take a very big number and check if it’s prime or no by dividing it. It’s easier to find the multiples of small prime numbers and check if they coïncide with the numbers we are checking on.
In addition, from (1) we already know x is 4,099 and from (2) we know that x is prime. So it’s a waste of time to check if it’s prime or no. 
What we care for, in (2), to know if it’s sufficient or no, is to know if there is another prime number between 4,095 and 4,105 other than 4,099.

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x is a positive integer such as 4,095<x<4,105. What is x? (1

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