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Primes from 11-109 (why you only need to memorize 49,77,91)

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
Expert replies
by scoowhoop » Mon Mar 08, 2010 7:38 pm
This is a little thing I put together, I haven't seen it anywhere on here so I figured I'd share....

In this method we use reverse logic by finding all the numbers that arn't prime and eliminating them. Since all of these primes (11-109) can only end in 1,3,7,or 9 we create a table for all possible outcomes. We notice a repeating pattern: (21, 51, 81), (33,63,93), (27,57,87), and (39,69,99) are all NOT prime and can be eliminated. This only leaves 49,77,and 91 as the other non-primes. Draw this matrix at the beginning of the test to help reference prime numbers. I've attached a better diagram if you have a hard time visualizing below. Hope this helps!

11 13 17 19
21 23 27 29
31 33 37 39
41 43 47 49
51 53 57 59
61 63 67 69
71 73 77 79
81 83 87 89
91 93 97 99
101 103 107 109

~Drew
Attachments
Primes from 11-109.JPG
Last edited by scoowhoop on Tue Mar 09, 2010 12:15 am, edited 1 time in total.
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Source: — Quantitative Reasoning |

by firdaus117 » Mon Mar 08, 2010 8:10 pm
Really helpful in quickly jotting down prime numbers.
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by Brian@VeritasPrep » Fri Mar 12, 2010 10:10 am
That's pretty neat, Drew - thanks for posting that!

Because my nature with regard to the GMAT is to always ask "why", I looked at this and my first inclination was to determine why that's the case. For anyone interested, it's this:

The pattern that Drew shows here with 21,51, 81; 33, 63, 93; 27, 57, 87; and 39, 69, 99 is that all of those are multiples of 3. By taking the base multiple of 3 (21, 33, 27, etc.) and adding 30s, you're just creating more multiples of 3 (3*x + 3*10 = 3 * (x+10).

Once those are out, and we've already taken out even numbers (anything ending in 2, 4, 6, 8, or 0) and multiples of 5 (anything ending in 5), we've already accounted for numbers divisible by prime factors 2, 3, and 5.

The next one up is 7, which brings us to 49 (7*7), 77 (7*11) and 91 (7*13). Essentially, these other numbers to "memorize" are the products of 7 and another prime number. Any other multiple of 7 has already been accounted for, as it's a multiple of 2, 3, or 5. And, in this range, we don't need to worry about any primes above 11, because 11^2 is 121; if that's the upper limit of our range, then any number larger than 11 would need to be multiplied by something less than 11 in order to be less than 121.


One other note on primes - I had a student once mention that she had been taught in school to "look for numbers next to multiples of 6" when seeking out primes. Why does that work? 6 is a multiple of the two smallest prime numbers, 2 and 3. So, any multiple of 6 won't be prime, and any number two away from a multiple of 6 will be a multiple of 2 (every second number - 2, 4, 6, 8, etc.) is a multiple of 2) and any number three away from a multiple of 6 will be a multiple of 3 (every third number - 3, 6, 9, 12, etc. is a multiple of 3). So, the only numbers that could be prime, once you get past 2 and 3, are one away from a multiple of 6: 5, 7, 11, 13, 17, 19, 23, 29, 31, 37...


To summarize this post (and thanks, readers, for indulging me), you can often learn the mechanics behind these shortcut tricks, and in doing so you'll internalize them that much more. If you ask yourself "why" when you see tricks and shortcuts posted here, you'll likely be able to apply those concepts even broader and more effectively on the GMAT.
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Chief Academic Officer
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