Brian Galvin is the Director of Academic Programs at Veritas Prep, where he oversees all of the company’s GMAT preparation courses.
For some of us, one of our favorite days in school was when our teacher let us use a calculator and, therefore, never had to draw out that feared long division bracket again. Long division is a meticulous, time consuming process and, as some of us tried to convince our grade school teachers, is a waste of paper given the numerous steps it requires a student to write down. So, when the GMAT asks us to conclude how many numbers in the range of 210 to 230 are prime, the thought that we may need to take numbers like 217 and divide them by 7 or 11 is frightening.
There is an easier way. If you note that a(b+c) = ab + ac (the distributive property), then you can also find that 7 * 31 = 7(30+1) = 7(30) + 7(1). Furthermore, if you want to determine whether 217 is divisible by 7, you can simply subtract a big multiple of 7 (like 210) to whittle away at the larger number 217. 217-210 leaves 7, which definitely divides by 7, so we know that 217 is divisible by 7.
If you follow that logic, you are close to having a fast method for determining divisibility, and to being able to quickly check numbers to see if they are prime. Let’s, again, address the question:
How many prime numbers exist between 210 and 230?
At first, you should recognize that you can eliminate all even numbers (they’re divisible by 2) and all numbers that end in 5 (they’re divisible by 5), leaving:
211
213
217
219
221
223
227
229
Next, test every number for divisibility by 3, using the method that, if the sum of the digits is a multiple of 3, then that number is divisible by 3. This works to eliminate:
213 (2+1+3 = 6) and 219 (2 +1+9 =12). Now we are left with:
211
217
221
223
227
229
From here, using divisibility “tricks” becomes trickier. While there do exist tricks for 7, 11, and other numbers, they tend to be time consuming and need extra memorization without much ROI. A simpler check for divisibility exists for all numbers:
Find a multiple of the number-in-question in the range that you are testing, and then just add/subtract the number-in-question from that to round up all of the multiples.
As we test this range for multiples of 7, the easiest multiple of 7 to note is 210. If 210 is a multiple of 7, then adding 7, and adding it again, will again produce multiples of 7:
210
217
224
At this point, we have already eliminated 210 and 224 (both even), so the only new number to eliminate is 217, leaving:
211
221
223
227
229
Next, we will test for divisibility by 11 using the same system. Our “anchor” will be 220 (the product of 11 and 20), which means that the next-closest multiples of 11 are 209 (220-11) and 231 (220+11), both of which fall outside the range in question. Here, we cannot eliminate any new values.
Next, test for divisibility by 13, again finding a number “in the range” and working from there. Here, it is easiest to use 260 as our “anchor”, and then work down to our range from there. Subtracting 13 from 260, we find that the following are multiples of 13:
247, 234, 221, 208
221 is in our range, so we can eliminate it, and we are left with the rest as prime numbers:
211, 223, 227, 229
Note that this approach also helps to avoid having to individually test each of the remaining numbers. If we know that they are not divisible by 2, 3, 5, 7, 11, or 13, we know that they are prime, as they cannot be divisible by any other number less than 16 (each of those numbers would be divisible by another in that set). Because 230, the upper limit of our range, is less than 162, then in order for a number in our range to be divisible by something greater than 16, that factor would have to be paired with something less than 16, and we have already tried each of those numbers. Therefore, we can conclude somewhat quickly that these four numbers are prime.
In conclusion, to avoid attempting several long-division calculations when checking numbers for divisibility, or for whether they are prime, merely find a multiple of a potential factor in the range of the number(s) you are testing, and use addition and subtraction to complete the set.
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