Hi,

The 30 Second Approach

P is a two digit number >0

Statement 1

P-2, P+2 is prime .

Considering P is two digit number P+2 will be prime in two digits. That means P+2 will be odd. This means P will be odd That means P-2 will also be odd prime .

Smallest double digit number is 10 and P-2 =8, this means P-2 is also double digit prime number .

Means P-2, P, P+2 is a series of Odd Numbers where at least one of the number is divisible by 3. That Means P( a double digit) will be divisible by 3 and hence is not Prime Since any double digit number divisible by 3 is never prime .

Statement 2:

P-4, P+4 is prime .

Considering P is two digit number P+4 will be prime in two digits. That means P+4 will be odd. This means P will be odd That means P-4 will also be odd prime .

Smallest double digit will be 10 and P-4 =6, ( P-4 can be odd prime >6) this means P-4 is not divisible by 3

Means P-4, P P+4 ( Actually a series of P-4,P-2,P,P+2,P+4) is a series of Odd Numbers where at least one of the number is divisible by 3.

Since P-4 is not divisible by 3 we have ( P-4+6 = P+2 is also not divisible by 3 & P+4 is not divisible by 3 we have ( P+4-6 = P-2 is also not divisible by 3

That Means P( a double digit) will be divisible by 3 and hence is not Prime Since both(P-4 and P+4) the prime will never be divisible by 3

Hence D.

However a Twist This question has a condition that leads us to answer . So what would happen if were not told that p is two digit number , how would we solve. Had we been only told that P is positive number then what would happen .Let's See.

Statement 1 tells us that P+2, P-2 are prime . Is P prime .

Things to note :

P-2, P , P+2

(1) Now observe the series P-2, P , P+2 its an arithmetic series with common difference 2. We cannot have P as even because , then P+2 and P-2 will also be even . And we are told that both (P-2 & P+2) are prime and there exists one and only one Even prime that is 2. So P is odd .

(2) That means we have a series of consecutive odd numbers, . Which means at least one of them will be divisible by 3.

(3) Indirectly the question is asking if P is one of the consecutive odd primes

(4) Min Value for which we need to check for the value of P would be 4 because all values of P below 4 would result into a negative number , 0 or not a prime . ( Smallest prime number is 2)

Drawing Conclusions from above we have

Now since we have a series P-2 being the smallest odd prime number in the series , if its not divisible by 3 ( this means its not equal to 3, Smallest Odd prime Number is 3 so series is off odd number greater than 3 ) , then for sure P+2 is also not divisible by 3, which means P is odd number greater than 3 and divisible by 3 . Hence we have P is not Prime.

But of P-2 is divisible by 3 which means we have a series of ( 3,5,7) in which case P is 5 and is prime

Hence Statement 1 is Insufficient we could also have following series (7,9,11) (13,15, 17) (19, 21, 23)..........(A, M, B)

Now Statement 2P-4 and P+4 are primes.

P-4, ..... P ...... P +4

Again P has to be odd . Since P+4, P-4 are Odd primes. where P-4 < P+4

Now we need to check for values Starting 7, since we are told that P-4 and P+4 are primes and we have deduced that they are odd prime ( i haven't heard of negative primes) any value below 7 would turn statement 2 either negative or 0. or not Odd prime

Again if P-4 is not divisible by 3 which means P-4 is not equal to 3 and is a prime greater than 3 then P+4 is definitely not equal to 3 Since P+4 is a prime . Which means

P+2 is also not divisible by 3 ( P-4+6 = P+2 ) , which implies P is divisible by 3 .Since P-4 is a prime greater than 3 and P is divisible by 3 we have P is not prime

However if P-4 is divisible by 3 which means P-4 = 3 then P=7 , we have P is prime

Now combining both statements together

P-4, P-2, P+2, P+4 are primes Is P prime .

Again P is odd and Minimum value we need to check is 7.

Again if P-4 is not divisible by 3 which means P-4 is not equal to 3 and is a prime greater than 3 then( P-2, P+2 P+4) is definitely not equal to 3 Since ( P-2, P+2 P+4) these are odd primes. We have P is number greater than 3 and is divisible by 3 . Which means P is not Prime

However if P-4 is divisible by 3 which means P-4 = 3 So the series would have been ( 3,5,7,9,11) , but notice that 9 is not prime in the series So P-4 can never be divisible by 3. Which Means P is divisible by 3 and is Odd number greater than 3 . Hence P is not Prime .

So Combining Both statement we have P is not Prime

An Important Learning

(a) Understand Why conditions are given ( at least on couple of problems , It will help you take in all possible scenarios)

(B) If there are Three consecutive odd numbers then one of them will definitely be divisible by 3

Hope this helps.

_________________

Probus

~You Just Can't beat the person who never gives up~ Babe Ruth