Here is a good problem to test your understanding of prime factors, not to add multiples. See if you can finish the problem in less than 2 minutes.

If P is the product of all of the positive multiples of 11 less than 100, then what is the sum of the distinct primes of P?

(A) 22

(B) 28

(C) 45

(D) 49

(E) 89

Approach: The first way to crack this problem is to figure out what the multiples of 11 are. Start with the lowest and begin writing them on your scratch paper. Doing so will help you map out the problem.

11, 22, 33… by the time you get to 33, you might want to stop and think what the range of the multiples is. Meaning, what is the highest multiple of 11 that is still less than 100. The answer is 99 = 11 X 9. Therefore there are 9 multiples of 11, starting with 11 and ending with 99.

The crux of the problem—at least in terms of saving time—is how to factor out the primes of P. Note that 11 x 22 x 33…x 99 is going to give you a really large number. How do we make it easy to look for the prime factors?

One way is to see which of the primes (2, 3, 5, 7, 11, 13 etc.) we can extract from 11 x 22 x 33…x 99. A quicker way is to factor out the 11 so that the multiples of 11, up until 99, can then be written as P = 11 (1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9).

Note that the problem asks us for the distinct primes, so we only need to pull one of each prime from P.

P = 11 (1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9).

Now we just need to add up 11 + 2 + 3 + 5 + 7 and we get 28. Answer (B).

These types of problems are time-consuming. Yet, there are always ways to make a problem less laborious. In the problem above, we looked for a pattern once we started writing out the numbers. Remember, if you are writing out all of the terms of the sequence that will cost you too much time. On other hand if you do not write out anything at all then you will not move forward on the problem. At the end of the day, the long way is better than the no way (unless of course you are running out of time.)

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Wow! I have never thought about these types of problems like that. I’ve definitely seen variations of these problems on the GRE and would always struggle with solving them. I’m a bit relieved to know that there are simpler and more time efficient ways of approaching these types of problems. Thanks for the insight!

I thought of a clever way to remember how to quickly get primes between 20 and 110. I did not include primes below 20 because I agree they should be memorized.

The general rule is In each 10s group there are two. The exceptions are 40 and 70, where there are three. This even follows a 2,2,3 pattern. The 90’s is an exception, there is only one, but the 100’s make up for it by having 4 (so the 2,2,3 pattern still works, and the only exception is 90).

Maybe that’s a lot to remember, but less than each of the primes. Then, simple testing of 2, 3, 5 and 7 eliminates non-primes.

Of course, 2 eliminates all even numbers. 5 eliminates all numbers ending in 5. Too easy to list.

Group Eliminate Primes

20-30 21 (/3), 27 (/3) 23, 29

30-40 33 (/3), 39 (/3) 31, 37

40-50 49 (/7) 41, 43, 47

50-60 51 (/3), 57 (/3) 53, 59

60-70 63 (/3), 69 (/3) 61, 67

70-80 77 (/7) 71, 73, 79

80-90 81 (/3), 87 (/3) 83, 89

90-100 91 (/7*), 93 (/3), 99 (/3) 97

100-110 none! 101, 103, 107, 109

* this is the only one that is not immediately obvious

Factoring out 11 from the product is specified incorrectly. It should have been as follows :

11*22*33*44…*99 = 11^9 (1*2*3*….*9)

Hye Chris

If we combine the prime factors of 2 numbers, will those represent the prime factors of their product?

For example

Combined factors of 8,5 and 10 are 2x2x2x23x5

Their product i.e 240 has identical prime factors.

Is this just an empirical observation or can we solve multiple questions on the GRE like this?

Hi Fatima,

Your approach above is definitely correct. I’m not sure how often a question on the GRE will be solvable with this exact technique. The main thing in the GRE quant is about how you reason with numbers and how are able to reason through problems that are similar–albeit slightly different–from previous problems.

One trap students fall into is they think they can apply the exact logic from one problem to another problem just because there is a superficial similarity between the two problems. The intuitive reasoning you show above is what should help you do really well on the quant section.

Let me know if that makes sense :)!

can you please demystify how come the combined factors of 8,5 and 10 gives 2*2*2*23*5

This is incorrect, 8*5*10=2^3*5*2*5=2^4*5^2 (=400 by the way). And this prime factorization is unique.

23 is a prime, not sure where it comes from.

The mentioned product is not 240 either, but 920. It seems that post is full of mistakes (or I’m missing many, many things).

I don’t get pretty much anything about the logic of that post (not yours).

yes that’s what even i was perplexed with !!

P=11*22*……*99=855652058110080

5+2=7

how 28?

Hi Shihan,

Mmm…that can kind of be confusing. The thing is the question is asking for the primes that make up P, not P itself (which would be a really big number). The only distinct primes that make up P, besides 11, are 2, 3, 5, 7. Their sum is 28.

Hope that helps!

Hi chris,

I understand the problem that was explained clearly upto adding 11+2+3+5+7 …U have mentioned as sum of distinct prime numbers ?? what is mean by distinct prime nums ?

Hi Sindu,

“Distinct primes” prime numbers are prime numbers that are different. To illustrate, if I asked you what are the prime factors of 36, you’d say 2, 2, 3, and 3. In other words, 2x2x3x3 = 36.

If I ask you what the distinct primes of 36 are, you would say ’2′ and ’3′. You don’t want to repeat the ’2′ or the ’3′, regardless of how many times they show up. So let’s say I asked you what the distinct primes of 96 are. 2x2x2x2x2x3. The distinct primes are still just ’2′ and ’3′.

Hope that helps!

Hi Chris,

I understand the problem right up until that last part where you can add up all of the distinct primes at the end. How does it work that we are able to just add up the primes to get the solution?

Hi Hana,

That’s because the question asks us for the “sum of the distinct primes.” That’s why we add up all the numbers.

Hope that helps clear things up!

Hey Chris,

Forgive my ignorance. What are distinct Primes?

Hi Harsha,

No worries :). All ‘distinct primes’ means is prime numbers that are different. To illustrate, if I asked you what are the prime factors of 36, you’d say 2, 2, 3, and 3. In other words, 2x2x3x3 = 36.

If I ask you what the distinct primes of 36 are, you would say ‘2’ and ‘3’. You don’t want to repeat the ‘2’ or the ‘3’, regardless of how many times they show up. So let’s say I asked you what the distinct primes of 96 are. 2x2x2x2x2x3. The distinct primes are still just ‘2’ and ‘3’.

Hope that clears things up!

Hi Chris,

Thanks a lot for the post. All your posts are very helpful 🙂

I have one quick question, how can you factor out 11 from the product of 11x22x33 …? You can factor out 11 when there is a sum 11+22+33+…

I mean, when you write

P = 11x22x33x44x55x66x77x88x99 = 855652058110080

But written as P=11(1x2x3x4x5x6x7x8x9) = 3991680

Which is not equivalent.

Perhaps I am missing something very obvious, because this is quite basic.

I am sorry if I asked a stupid question.

Cheers,

Rafal

I thought the same… You cannot really factor out a number from a product! but what you could do is to write the product this way:

(11*1)*(11*2)*(11*3)*…*(11*9), and this way you can clearly see which distinct prime numbers are involved in this product: 11, 2, 3, 5, 7. If you add them up you get 28, which is the correct answer.

Yes, that is a good point! Factor was the wrong word :). Separating the numbers from the product–the way that you did above–is more in line with what I was thinking.

Sorry for any confusion :).