A 20,000 dollar question!

Congratulations to Daniel Mai for winning the 2019 Raytheon MATHCOUNTS National Competition! Here's the question that sealed his victory.

Source
ESPN webcast of 2019 MATHCOUNTS National Competition (I can view it now, but videos like this sometimes expire and are viewable in specific geographies--so keep that in mind if the link does not work)

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[MindYourDecisions]

And I totally forgot 5040 = 7! = 2 x 3 x 4 x 5 x 6 x 7, which a few people nicely shared in the comments! We can remove the unique prime factors 2, 3, 5, and 7, leaving 4 and 6, so then 4x6 = 24. In a competition where seconds matter, this is a much better way to solve it!





*I hope you can see the webcast, but sometimes the videos expire and they are licensed only to be watched in certain countries. If the link doesn't work, be patient as the competition sometimes gets posted officially to YouTube too.

[vlahovivan]

5040 = 7!, so only two non prime numbers in the product are 4 and 6, which when multiplied give 24

[kelperdude]

I had to google the word quotient before I could start.

[dp]

It took me around five seconds. I immediately recognized 5040 as 7! (2×3×4×5×6×7). Unique prime factors are 2, 3, 5 and 7. Left are 4 and 6, which multiplied (4×6) gives the answer, 24.

[GAshoneybear]

Aww, man. I misread the question thinking the unique prime factors meant it only occurred once. So I got 144.

[BigParadox]

I solved it in a few seconds in my mind. I remembered that that 5040 is 7 factorial (7!), which is 1*2*3*4*5*6*7. I see that of those numbers 2, 3, 5 and 7 are primes. The numbers left out are 4 and 6, but they do not have any other primefactors than what we already saw, so the unique prime factors are only 2, 3, 5 and 7, and thus, if you divide 5040 by that, you will of course get 4*6 = 24.

[DamienConcordel]

All you need to do is recognize that 5040 = 7! = 1 × 2 × 3 × 4 × 5 × 6 × 7. Remove 2, 3, 5 and 7 and you're left with 1 × 4 × 6 = 24. That's probably how he got it in seconds.

[pedroejenny]

I didn't recognize 5040 as 7!, but solved it by only dividing once for each prime. So from 5040, I divided by 2 to get 2520 than by 3 to get 840, by 5 to get 168 and finally by 7, to get 24. As 24 can only be devided by 2 and 3, that is the answer.

[demwz]

I was able to solve it in seconds, but i missed the therm "uniqe"
If you devide something by the product of its prime factors you get: 1

[spacelightning]

The only part I feel shaky on is our definition of "unique." Like, we treated it to mean each prime factor only once, but I feel removing any non-unique one entirely would also be valid, resulting in 144

[crtwrght]

I didn't spot the factorial (obvious in retrospect) but did happen to recall that there are 10080 minutes in a week (don't ask). Not as elegant but still a fairly quick solve to sort out 60 * 12 * 7 (instead of 60 * 24 * 7 for the week).

[prasundutta]

One of the very few questions of the channel which I was able to solve

[EllipticGeometry]

My ‘shortcut’ was to remember that 2520 is the LCM of positive integers up to 10. 5040 is twice that. That implies our primes are 2*3*5*7=210. Divide by that to make 24.

It sure would’ve been easier if I recognized factorials beyond 6! on sight.

[cpsof]

You don't have to do the prime factorization. Just divide by each prime number as long as possible. 5040 / 2 = 2520, 2520 / 3 = 840, 840 / 5 = 168, 168 / 7 = 24. Since 24 is not divisible by a prime number bigger than 7, we are done.

[Tiqerboy]

The way I solved it was a bit different than your video:

5040 / (2 x 5)
= 504 / 3
= 168 / 7
= 24

And we can't go any further since we have already used 2 and 3, the prime factors of 24, so that is the answer.

[borstenpinsel]

Did you not read the word "unique"? 2 is not a unique prime factor to the number. The prime factors are 2, 2, 2, 2, 3, 3, 5, 7 and only 5 and 7 are unique. questions like these, where the answer is not related to the question at all gives students the impression they don't get math.

[Rekko]

I think the answer should be 144 because 5 times 7 is 35, and when you divide 5040 by 36 you get 144. 2 and 3 are not unique prime factors because they are there more than 1 time. This question is very misleading.

[dabluetoad]

I was actually in the Texas State math counts competition, and it was really fun, and I got absolutely crushed. Still a cool experience though!

[preritgoyal]

My approach was to divide by 10 as 2, 5 are its prime factors.
Then 504 is left and primes 3, 7 divide it. So dividing by 21 gives 24.
And no primes other than these divide 24.
So 24 is the ans.

[JohnDlugosz]

I did it in seconds in my head. I recognize that 5040 is the factorial of 7. Reading too quickly "divide by (all) factors" gives 1 -- easy trick. But then I saw "unique factors" and took the prime numbers <=7 as them. What's left? 4 and 6. The product of those will be the answer.