x and y are prime numbers such that 9

Question

x and y are prime numbers such that 9 < x < 100 and y is the reverse of the digits of x, what is the value of xy?

(1) x - y = 18

(2) x + y = 44

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lacktutor · Answer

x and y are prime numbers such that 9 < x < 100 and y is the reverse of the digits of x, what is the value of xy?

(Statement1): x —y = 18
x = 10a+ b
y= 10b+a

x —y = 10a+ b—10b—a=9a—9b=18
—> a —b= 2

31–13= 18
97 —79= 18
Insufficient

(Statement2): x+ y = 44
10a +b +10b+a= 11a +11b= 44
a+ b= 4
—> 3 + 1= 4
—> 1+3=4
31+13= 44
13+31=44
x*y= 31*13= 13*31
Sufficient

The answer is B

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Archit3110 · Answer

given x & y are prime no
#1
10x+y-10y-x=18
x-y=2
we can have (31,13) '; ( 35,53) ; ( 79,97)
insufficient
#2
x+y=44
x+y=4
13, 31 
sufficient
IMO B

ccooley · Answer

There are some serious fireworks going on in the question stem! Let's start there.

We know that x is between 9 and 100. That's a complicated way of saying that x is a 2-digit number. So, y will also be a 2-digit number. x and y are the reverse of each other, and x and y are both prime.

My rule of thumb is that  if I think there are 10 or fewer total possibilities that match the constraints, I'll just write them all down before I start.  I think that's probably the case here. How many prime numbers can there possibly be where the reverse is ALSO prime? Probably not too many, right?

Just to be safe, I'll reduce the possibilities a little bit more so I don't need to spend as much time looking. I don't need to look at any prime numbers that start with an even digit, since their opposite will be even, and therefore it won't be prime. I also don't need to look at any prime numbers that start with 5, since the opposite will be a multiple of 5, and won't be prime.

So I only need to think about prime numbers that start with 1, 3, 7, or 9. I can jot those down pretty fast:

11, 13, 17, 19
31, 37
71, 73, 79
97

Now, which ones are still primes when they're reversed? Here are the possibilities:

11, 11
13, 31
17, 71
37, 73
79, 97

Five possibilities. I was right, there are fewer than 10!

Each of these pairs will have a different product, and the question asks for the product. So, a statement will be sufficient  if it lets us narrow things down to exactly one of these pairs.  I'd rather add than subtract, so I'll start with statement 2.

Statement 2:  x + y = 44. There's only one pair up there that sums to 44, and that's 13 and 31. (It doesn't matter which is x and which is y, since the product would be the same either way, and that's what we're trying to solve for.) So, this statement is sufficient. Eliminate A, C, and E.

Statement 1:  x - y = 18. Yikes - I see two pairs that have a difference of 18. 13 and 31 works, but so does 79 and 97. So, I can't narrow it down to just one pair. Therefore, this statement is insufficient.

The answer is B.

Samrock · Answer

Can someone plz explain how x=10a+b and y=10b+a

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x and y are prime numbers such that 9 < x < 100 and y is the reverse

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