FPD´s 6th Edition Chap 9 Problem Set Question 7

[JoaoM763]

Hello,

This question (FPD´s 6th Edition Chap 9 Problem Set Question 7) uses symbols instead of numbers. Its stated that the numbers must multiply to 36 and because of this the solution assumes that 2,3 and 6 must be the numbers. The option considering 1,4 and 9 are discarded.
I would like to understand what rational was used to discard any combination using 1,4 and 9 right away.

Tks in advance.

[RonPurewal]

hello,
per the forum rules, please provide the complete text of the problem as originally written. if there are answer choices, please include those, too.

thanks.

[LiliiaG24]

Hello, I have the same question. Why can't we use 4 and 9, since the prime numbers are 2,2,3,3.
The complete question is below:

In the multiplication above, each symbol represents a different unknown digit, and round x square x rhombus =36. What is the three-digit integer round, square, rhombus?
A) 263 B) 236 C) 194 D) 491 E) 452

Round rhombus
X
Square rhombus
______________________
Triangle square rhombus

Thank you!

[RonPurewal]

ah okay.

you can't use 1,4, and 9 because they don't work.

remember—your three digits have to multiply to 36 AND they have to make the given multiplication problem work.
with the digits 1, 4, and 9, there's no way to get the multiplication to work out like that.

[RonPurewal]

also, note that you can just BACKSOLVE this problem.

each answer choice gives you the 'round', the 'square', AND the 'rhombus'. that's a complete set of information—there's nothing left to figure out. you can just try each set and see whether the multiplication works.

choice A gives 23 x 63. this does not have the form _63.
choice B gives 26 x 36. this is 936. this works (it's _36).
you're actually DONE here, since there will not be more than one correct answer.

but, if you check the choices with your digits, you'll find that they don't work anyway:
C would be 14 x 94, which ends with '6' and so can't possibly work.
D would be 41 x 91, which is way way waaaaayyy more than 1000 (it's more than 40 x 90 = 3600) and so is not a three-digit number.

so there's nothing wrong with the digits 1, 4, 9 per se; they just don't do what they have to do IN THIS SPECIFIC PROBLEM.

[RonPurewal]

make sure you don't neglect the backsolving solution, by the way. it's certainly more straightforward than using primes, and possibly more efficient as well.

[LiliiaG24]

Backsolving is much easier indeed. Thank you Ron!

[RonPurewal]

Backsolving is much easier indeed. Thank you Ron!

...and, indeed, a great many problems are specifically engineered that way.

remember—the official (OG) solutions won't tell you about things like backsolving and smart numbers—but, think about it: let's say i have a retail store, and i'm willing to haggle/bargain with my customers. even if i'm willing to haggle WAY down on the prices, there's still no way i'm going to hang out a sign saying, 'hey! i'm willing to bargain!'
...because of course i'd be glad to let people come in and pay full price all the time, if they don't know any better.

in this analogy, 'paying full price' = using 'textbook' methods all the time.
'not hanging a sign...' = not giving away the game, so that people who actually know how to play the game retain an advantage over those who don't.