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## FAQ

Can you add 5 odd numbers to get 30?

It is 7,9 + 9,1 + 1 + 3 + 9 = 30Wish you can find the 7,9 and 9,1 in the list of1,3,5,Â Â 7,9Â Â Â ,11,13,151,3,5,7,Â Â Â Â Â 9,1Â Â Â 1,13,15

How many colleges I can fill out as choices for the All India 15% Quota?

There isnâ€™t any limit as to how many choices you can fill in during the counseling. In the end remember to put those choices at the top which you prefer over others. Seats are allotted based on your preference so choose wisely.

How would the SF Bay Area be different today if they'd used some of the gold-rush money to build a dike, reroute the rivers, pump out the water and fill it with dirt up to ground level?

So many problems with this question.First: Scale. Do you have any idea HOW LARGE the SF Bay is? 1,600 Square Miles! How on earth do you fill all that in?Second: Flooding. The bay is there because three major rivers (Sacramento, American and San Joaquin) come together here. That plus innumerable other smaller waterways means that you have millions of gallons of water coming into the bay every day. Where would all that water go?Third: The port. The real reason why San Francisco exists is because of the bay. There is really nothing terribly interesting about San Francisco during the gold rush. Except for the fact that it was on the way to the gold fields. The Bay made this possible. In this time period, water was the major transportation mode. Filling in the bay would have been a disaster for them economically.Fourth: Environmental impact. There are several major ecosystems represented by the bay. We did huge damage to them during the first part of the 20th century and are just starting to set them right again.Fifth: Aesthetics. The bay is beautiful and our lives revolve around it. Why on earth would we fill it in?You get the idea.Filling in the bay would have pretty much ended San Francisco as we know it. Perhaps the city would not be there at all.BTW, the opposite of what you ask actually happened. During the ice age you could walk to the Farallons because the sea level was so much lower:

Mathematical Puzzles: What is + + = 30 using 1,3,5,7,9,11,13,15?

My question had been merged with another one and as a result, I have added the previous answer to the present one. Hopefully this provides a clearer explanation. Just using the numbers given there, it's not possible, because odd + odd = even, even + odd = odd. 30 is an even number, the answer of 3 odd numbers must be odd, it's a contradiction. If what people say is true, then the question is wrongly phrased its any number of operations within those three brackets must lead to 30. Then it becomes a lot easier. Such as 15 + 7 + (7 + 1). That would give 30. But it assumes something that the question does not state explicitly and cannot be done that way. I still stick to my first point, it can't be done within the realm of math and just using three numbers, if not, then the latter is a way to solve it.EDIT:Â Â This question has come up many times, Any odd number can be expressed as the following, Let [math]n, m, p[/math] be an odd number, [math] n = 1 (mod[/math] [math]2), m = 1 (mod[/math] [math]2), p = 1 (mod[/math] [math]2)[/math][math]n+m+p = 1 + 1 + 1 (mod[/math] [math]2)[/math]Let's call [math]n+m+p[/math] as [math]x[/math][math]= x = 3 (mod[/math] [math]2)[/math]Numbers in modulo n can be added, I'll write a small proof for it below, [math]a = b (mod[/math] [math]n), c = d (mod[/math] [math]n)[/math][math]a+c = b+d (mod[/math] [math]n)[/math]We can rewrite [math]b[/math] and [math]d[/math] in the following way, [math]n | (b - a) = b-a = n*p[/math] (for some integer p) [math]b = a + np[/math][math]b = a + np, d = c + nq[/math][math]b + d = a + np + c + nq[/math][math]b+d = a + c + n(p + q)[/math]Now we have shown that our result is true, moving forward, [math]3 = 1 (mod[/math] [math]2)[/math][math]x = 1 (mod[/math] [math]2)[/math]Therefore the sum of three odd numbers can never be even. It will always be congruent to 1 in mod 2.(This was what I wrote for a merged answer).Modular arithmeticÂ - Link on modular arithmetic, the basic operations. Modular multiplicative inverse - The multiplicative inverse in modular operations.Congruence relationFermat's little theorem Modular exponentiation - As title suggests.Good luck!

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