In the multiplication above, each symbol represents

[aliaskjw]

Hello,

My question concerns Fractions, Decimals & Percents, 6th Ed, p. 125, Question 7. The question uses symbols that this forum's character set does not appear to support, so rather than slowing things down with an image, I am replacing the symbols with the variables A, B, C and D. (It doesn't change the problem.)

The question and possible answers:

7)

A B
x C B
----------
DCB

In the multiplication above, each symbol [in this case, letter] represents a different unknown digit, and A x C x B = 36. What is the three digit integer ACB?

A) 263 B) 236 C) 194 D) 491 E) 452

My Confusion:
On p. 128, the book discusses why the answer is B (236). I understand each step of the explanation and the overall explanation, and how the author arrived at the answer. What I don't know is how they arrived at one of the steps along the way. In particular: (quoting)

The three symbols [letters] of ACB multiply to 36 and each must represent a different digit. Break 36 into its primes: 2 x 2 x 3 x 3. What three different digits can you create using two 2's and two 3's? 2, 3, and 6.

My question: how did the author (and how would a test taker on exam day) know to break down 36 into its prime factors? I know it's stated that the product of ACB is 36 in the problem, but performing a prime factorization (as a method of narrowing down the possible individual digits) didn't naturally occur to me. What is the clue that this prime factorization needs to be done in order to solve the problem?

Thank you for any help anyone can provide!



Kirk

[Chelsey Cooley]

You don't technically need prime factorization to solve this one. You're trying to answer the question "what is a set of three unique one-digit numbers that you can multiply together to come up with 36?". One way to do it without factorization would be to start with the digit 1 and then work your way upwards, testing each possibility:

36 = 1 x 2 x 18 (bad case, because 18 isn't a digit)
36 = 1 x 3 x 12 (bad case, because 12 isn't a digit)
36 = 1 x 6 x 6 (bad case, because C and B would both represent the same digit)
36 = 2 x 3 x 6 (hooray!)
etc.

The takeaway there is that prime factorization isn't a magic bullet. It's just one of several possible ways to answer a question that asks you for the 'pieces' that multiply together to make a larger number. (By the way, that's my answer to your question - when a problem wants you to figure out which integers multiply together to make another, bigger integer, then finding the factorization of the bigger integer is a good starting point. Factorization is really just a tool for finding the 'smaller pieces' inside of a larger number.)

[RonPurewal]

...or you could just backsolve.

you're given that the answer represents 'ACB', so this is THE best possible situation for backsolving: each answer choice contains ALL of the digits 'A', 'B', 'C'!

...so, from each answer choice, you can just form 'AC' and 'BC'. then you can just multiply them together, and see whether the result has the desired form (= it must end with the same two digits that are already designated 'BC').

choice A gives 23 x 63.
this is __9 (there's no point in doing anything past this point, since the last digit is not 3).
nope.

choice B gives 26 x 36.
this is 936, which has the requisite form.

...done.

[RonPurewal]

also note that you can narrow the choices down to B and D very quickly, just by looking at the ones digits of the numbers (= the first things to multiply, if you're performing traditional long-hand multiplication).

for choices A, C, and E, the right-hand column doesn't work. (for choice A, _3 x _3 is not __3... and so on.)

for B and D the right-hand column works, so it's necessary to work those two a bit more. but only those two.