The factors of 434 are listed below the puzzle, but here is a quick problem (really a puzzle) for you to figure out:

434 is the sum of six consecutive prime numbers. One of those primes is 71. What are the other five? If you would like, you may type your answer in the comments.

Solving that problem will require you to work backwards.

Many students may think they understand a mathematical concept, but they don’t really understand it unless they can also do it backwards. For example, they may understand, but maybe not remember that 7 x 8 = 56. Yet, these same students are completely baffled if you tell them this rhyme, “Five, six, seven, eight, 56 is 7 times 8.” They aren’t used to seeing addition, subtraction, or multiplication facts written or spoken in that backwards way. Coleen Young made an excellent slide presentation titled good-mathematicians-can-go-backwards. We ALL can go forward in mathematics so much more confidently once we learn to go backwards.

Coleen recommends many resources including the Find The Factors puzzles. Each of these puzzles can be solved and then completely filled out as a multiplication table if you will go backwards first and find the factors of the clues in the grid. Think what two numbers multiplied together give you 49? Give it a try!

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-23

- 434 is a composite number.
- Prime factorization: 434 = 2 x 7 x 31
- The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 434 has exactly 8 factors.
- Factors of 434: 1, 2, 7, 14, 31, 62, 217, 434
- Factor pairs: 434 = 1 x 434, 2 x 217, 7 x 62, or 14 x 31
- 434 has no square factors that allow its square root to be simplified. √434 ≈ 20.8327

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I think either of your clues gave too much in the way of a clue. 71 immediately tells me to look at primes in the …, 67, 71, 73, … range, and telling me six primes are involved again points me towards a prime close to 434÷6 being in the middle of the six. But if all I know is that a group of successive primes is involved I’ve got to work a bit harder (so please set us one at the first opportunity!)

You’re saying you would prefer not to be given any of the prime numbers nor the number of consecutive primes?

I think so. Otherwise the clues are just too helpful. Maybe the puzzles won’t be all that interesting anyway, but without any clues at all I won’t necessarily know if I’m looking for an odd or even number of primes.

Unless one of the consecutive primes is 2, if the number is even, it must have an even number of consecutive primes. If the number is odd, it must have an odd number.

Would it be okay to give this clue: the number of primes is between 1 and 10?

Try it and let’s find out! As you point out, if 2 is not in the picture I can immediately eliminate 50% of the possibilities I need to look at, so that’s one reason for limiting the amount of information you should give.

I sometimes joke that to completely understand a math concept, you should be able to do it forward, backward, sideways or from any direction 🙂

Since math is perceived to be a rigid subject, many assumed that you can’t use different methods in solving a problem but of course, that’s not the case. There are problems especially in number theory and combinatorics that can be solved in several ways from several directions after all.

You are absolutely right, and as you well know, some methods almost work like magic!

Since the introduction of calculators, many students can’t recognize 81=9×9 mentally! Good point about the backwards thinking.

Thank you. Too much dependence on calculators does make sense. I’ve seen 14 year olds who couldn’t divide 18 by 2 without one. I haven’t seen very many 8 or 9 years old using them, however. I think never seeing or hearing backward equations like 72=8×9 makes a difference, too.

Just curious, what software do you use to make the puzzle?

I make the puzzles by hand. I type them into excel to make them look good, and use paint to make them jpg’s so they look good in wordpress.

I see! Excel is really useful.