<p>1- For all positive integers a, let [a] be defined as the sum of the odd integers from 0 to a, inclusive. For example, [8] = 1+3+5+7=16. what is the value of [100] - [96]? </p>
<p>2- How many positive integers less that 1000 are NOT divisible by 3? </p>
<p>So is there any way for these kinds of questions that I can use to get to the answer quickly?</p>
<p>For the second one it says How many positive integers LESS that 1000 are NOT divisible by 3
so use divide 999 by 3 999/3=333
333 is not the answer 333 is all the numbers that are divisible by 3
so we subtract 333 from 999 which is 999-333= 666</p>
<p>The answer to the first one is 99 + 97, or 196.</p>
<p>[100] is a lot more than that, of course, but you don’t need to compute [100] or [96]. You just have to recognize that when you subtract [96] from [100], all the odd integers from 1 through 95 in [100] will be subtracted out by [96], leaving only 97 and 99.</p>
<p>It’s akin to dividing out factorials. It’s easy to find the answer to (25!)/(23!), even though 25! and 23! are huge numbers. But the positive integers from 1 through 23 in 25! get canceled out by the 23! in the denominator, so (25!)/(23!) = 24*25 = 600.</p>
<p>I think the for the first one [100]=2460 and [96]=2264
[100]-[96]=2460-2264= 196</p>
<p>Thank you both! Wow I didn’t realize that the first question is so blatant and easy!</p>
<p>Could be, Misanthropic. The numbers seem right, but I didn’t bother to compute them.</p>
<p>And if you’re doing a question such as this on a timed test, I think actually computing [100] and [96], even if you can take short cuts by summing series, would be a less-than-optimal of time if you can find a faster way.</p>
<p>Technically, [100] = 50^2 = 2500 and [96] = 48^2 = 2304. (not 2460, 2264). In either case we (somehow) obtain the correct answer.</p>
<p>But, in order to prevent making errors, we note that [100] - [96] is just 97+99 = 196.</p>
<p>rspence, you computed the sum of ALL the numbers from 0 to 100, but the question asks for the sum of the “odd” numbers, so it’s 2460.</p>
<p>rspence, you computed the sum of ALL the numbers from 0 to 100, but the question asks for the sum of the “odd” numbers, so it’s 2460 not 2500. Am I right?</p>
<p>Actually, the sum of all the whole numbers from 0 through 100 is 5050, and the sum of the whole numbers from 0 through 99 is 4950.</p>
<p>The sum of the odds from 1 through 99 is indeed 2500. It’s (1 + 99) + (3 + 97) + (5 + 95) + … + (49 + 51). Each of those pairs totals 100; there are 25 such pairs.</p>
<p>I see! Sorry it’s my mistake.</p>