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What Number am I? – Square, cube, prime digits

Math Teaser
I am a 3 digit number whose product of my digits is equal to 144. Of my 3 digits, one is a cube, one is a square, and one is a prime. If the six numbers that meet these criteria are listed in increasing order, I am fifth in the list.

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To solve mathematically, start by factoring 144 to get the basic multiples – 2 x 2 x 2 x 2 x 3 x 3. These values can help you find the basic combinations of digits that multiply to equal 144. Any combination that multiplies to a single digit value can be the value of one of the digits (that then multiply to equal 144) and the remaining multiples can be used in the same way to find the other digits. If you found every combination that produced 144, you could write a list of the 3 digit numbers – there are 18 such combinations of 2,3,4,6,8,9 – and you could then use this list to find the combination that meets the criteria of the digits.

In this problem, it may be more direct to analyze the criteria. One digit is a cube – there are only two single digit cubes – 1 (1 x 1 x 1) and 8 (2 x 2 x 2). One digit is a square – there are only three single digit squares – 1 (1 x 1), 4 (2 x 2), and 9 (3 x 3). One digit is a prime – the single digit primes are 1, 2, 3, 5, and 7. So which combination makes sense? Let’s start narrowing it down. Anything with a 1 doesn’t belong because that would leave the other two digits to multiply to 144, and that can’t happen. It should be obvious that 144 can’t be a product of anything multiplied by a 5, so the 5 is out. What is left? For the cube digit, with the 1 gone, only the 8 is left – we have found one of the digits. For the square digit, it could be the 9 or the 4, and it is simple to test those choices with the square and the remaining prime digits (2,3,7) to see which multiply to equal 144. With some trial and error, you can see that the square digit is 9, the cube digit is 8, and the prime is 2 – all multiplying together to equal 144. Put the six possible combinations of these digits in increasing order (289,298,829,892,928,982) and count to the fifth in the list – 928 is the answer.