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A three digit number is such that its hundredth digit is equal to the 14 Jun 2021, 07:45
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A three digit number is such that its hundreds digit is equal to the product of the other two digits which are prime numbers. Also, the difference between the number and its reverse is 297. Then, what is the ten's digit of the number?

(A) 2
(B) 3
(C) 5
(D) 6
(E) 7
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A
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A three digit number is such that its hundredth digit is equal to the 14 Jun 2021, 08:01
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I'm guessing a bit what the question means, because the standard of writing is poor, but if the tens digit and units digit are both prime, and they multiply to the hundreds digit, then the tens and units digits can only be 2 or 3 (since if either was 5 or larger, we'd never get a product less than 10). I suppose you could now list the small number of possibilities: 422, 623, 632 and 933 and test each to determine which is correct. But if we take a three digit number ABC, and reverse its digits to get CBA, then if these numbers differ by roughly 300, then the digits A and C will differ by 3. So the number must be 623, and the tens digit must be 2.

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A three digit number is such that its hundredth digit is equal to the 14 Jun 2021, 23:08
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Bunuel
A three digit number is such that its hundredth digit is equal to the product of the other two digits which are prime numbers. Also, the difference between the number and its reverse is 297. Then, what is the ten's digit of the number?

(A) 2
(B) 3
(C) 5
(D) 6
(E) 7
Show more

If anyone wants to solve this algebraically:
Let the three digits be x(hundredths), y(tenths) and z(units) respectively.
So, number = 100x + 10y + z
Given: x = yz - eq(1)
Also, given:
100x + 10y + z - (100z + 10y + x) = 297
100x + 10y + z - 100z - 10y - x = 297
99x - 99z = 297
x - z = 3
From eq(1):
yz - z = 3
z(y - 1) = 3
The only possible combination is y = 2, z = 3(As y and z are prime numbers)
y = 2, IMO(A)!
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A three digit number is such that its hundredth digit is equal to the 15 Jun 2021, 01:12
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Let a three digit number= PQR or more precisely 100P + 10Q + R

=> Hundred's digit (P) = Q * R [Q and R are prime numbers]

The only possible values of Q and R are (2, 3) or(3, 2) as P is a single-digit number. So, Q = 2 and R = 3 or Q = 3 and R = 2 and P = 6 in both case.

=> Difference between the number and its reverse is 297: 100P + 10Q + R - (100R + 10Q + P) = 297

=> 99P - 99R = 297

=> 11P - 11R = 33

=> 66 - 33 = 11R

=> R = 3 and hence Q = 2.

Therefore, ten's digit = Q = 2.

Answer A
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A three digit number is such that its hundredth digit is equal to the 17 Feb 2024, 02:50
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I'm guessing a bit what the question means, because the standard of writing is poor, but if the tens digit and units digit are both prime, and they multiply to the hundreds digit, then the tens and units digits can only be 2 or 3 (since if either was 5 or larger, we'd never get a product less than 10). I suppose you could now list the small number of possibilities: 422, 623, 632 and 933 and test each to determine which is correct. But if we take a three digit number ABC, and reverse its digits to get CBA, then if these numbers differ by roughly 300, then the digits A and C will differ by 3. So the number must be 623, and the tens digit must be 2.

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­I Ian thank you very much for sharing your reasoning! It helped my a lot.
What I do not understand is when you say "reverse its digits to get CBA, then these numbers differ by roughly 300, then the digits A and C will differ buy three".
If CBA = 297, then ABC would = 792 which does not make any sense. None of the foud alternatives reverse to 297.


When I first read the question I understood the "reverse" number as the reciprocal.... Is this a huge misunderstanding? Or is it that the wording is confusing? I am not a native English speaker.
:)­
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A three digit number is such that its hundredth digit is equal to the 17 Feb 2024, 10:57
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ruis
IanStewart
I'm guessing a bit what the question means, because the standard of writing is poor, but if the tens digit and units digit are both prime, and they multiply to the hundreds digit, then the tens and units digits can only be 2 or 3 (since if either was 5 or larger, we'd never get a product less than 10). I suppose you could now list the small number of possibilities: 422, 623, 632 and 933 and test each to determine which is correct. But if we take a three digit number ABC, and reverse its digits to get CBA, then if these numbers differ by roughly 300, then the digits A and C will differ by 3. So the number must be 623, and the tens digit must be 2.

Posted from my mobile device
­I Ian thank you very much for sharing your reasoning! It helped my a lot.
What I do not understand is when you say "reverse its digits to get CBA, then these numbers differ by roughly 300, then the digits A and C will differ buy three".
If CBA = 297, then ABC would = 792 which does not make any sense. None of the foud alternatives reverse to 297.


When I first read the question I understood the "reverse" number as the reciprocal.... Is this a huge misunderstanding? Or is it that the wording is confusing? I am not a native English speaker.
:)­
Show more


297 isn't either of the two numbers.

297 is the DIFFERENCE between the two numbers.

One of the two numbers could be called ABC, where A is hundreds place, B tens place and C the ones place.

The reverse of this number ABC means to create a new number CBA.

So the difference between these two numbers is:

ABC - CBA = 297

To facilitate this subtraction these numbers can be rewritten as:

(100A+10B+C) - (100C+10B+A)

which can be simplified to:

99(A-C) = 297.

Dividing through by 99:

A-C = 3 or A=C+3

So, because C is prime and A must be less than 10, these are the only choices for the pair:

A C
5 2
6 3
8 5

Now, it's further stipulated that A = B*C

So looking at the above options:

Can A=5 ?

5=B×2 NO B works

Can A=6 ?

6=B*3 YES. B=2

Can A=8 ?

8=B*5 NO B works

So, the only number that works is A=6 B=2 C=3

623
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