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?
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Wait, the difference between the number and its reverse is 297? That's huge. For a 3-digit number and its reverse, the difference is always 99 times the difference between the hundreds and units digits, right? So 297/99 = 3. So the difference between the hundreds and units digits is 3. But then the hundreds digit is the product of the other two digits, which are prime... I'm getting tangled. Can someone explain the logic step by step?
Yeah, the difference is 99*(hundreds - units) if hundreds > units. So 297/99 = 3. So hundreds - units = 3. Then let the tens digit be t and units be u. Hundreds = t*u. Also t and u are prime. So t*u - u = 3 => u(t - 1) = 3. Since u is prime, u can be 3 and t-1=1 => t=2. Or u=1? But 1 is not prime. So t=2. So ten's digit is 2.
Thanks, that makes sense. I was overcomplicating it. So the key is that the difference between a 3-digit number and its reverse is 99 times the difference of the hundreds and units digits. Then it's just a simple equation.
I got ten's digit = 2. But let me double-check: if ten's digit is 2 and units is 3, then hundreds = 6. Number is 623. Reverse is 326. Difference is 297. Works. So answer is 2. But wait, could units be something else? If u=3, t=2, that's the only prime solution. If u= something else? u(t-1)=3, so u=1 or 3. 1 not prime. So only u=3, t=2. So ten's digit is 2. Answer A.
This one took me a while. The prime condition is the tricky part. I initially forgot that 1 is not prime. So I got stuck with u=1. Once I remembered that, it was straightforward. Good question.
Is there a faster way to do this? I plugged in answer choices for ten's digit and checked which one works with the prime condition and the difference. That took less than a minute. But I'm curious about the algebraic method.
Algebraic is faster: let tens = t, units = u. Hundreds = t*u. Difference = 99*(t*u - u) = 297 => t*u - u = 3 => u(t-1)=3. Since u prime, u=3, t-1=1 => t=2. Done.
What it tests
Your command of divisibility, primes, factors, parity, and how the GMAT tests properties of integers.
Common trap
Forgetting that 1 is not prime, that 0 is divisible by every integer, or that negative numbers flip your assumptions.