About Mathematics

Prime Generating Polynomials : The polynomial,  n 2 + n + 41 n 2 + n + 41  can be used to produce 40 primes for consecutive integer values 0≤n≤39.  This property was discovered by Euler . Similarly, the incredible formula,  n 2 − 79 n + 1601 n 2 − 79 n + 1601  was discovered, which produces 80 primes for the consecutive values 0≤n≤79! Kaprekar’s Constant : 6174 is known as Kaprekar’s Constant , after the Indian Mathematician D.R. Kaprekar. Take any four-digit number, using at least two different digits (Leading zeros are allowed.) Arrange the digits in descending and then in ascending order, to get two four-digit numbers, adding leading zeroes if necessary. Subtract the smaller number from the larger number and go back to step 2. The above process will always reach the fixed number, 6174, taking at most 7 iterations. Try it yourself! ( Note that after the first subtraction or the subsequent subtractions, the result obtained is always a multiple of 9!) Collatz Con

How is tan (A+B) = [tan A + tan B]/ [1 - tan A tan B]? Method 1: Let A = 30 deg and B = 45 deg. LHS = tan (30+45) = tan 75 = 3.732050808 RHS = [tan A + tan B]/[1 - tan A tan B] = [tan 30 + tan 45]/[1 - tan 30 tan 45] = [0.577350269 + 1]/[1 - 0.577350269*1] = 1.577350269/0.42264973 = 3.732050808 = LHS Proved. Method 2: tan (A+B) = [tan A + tan B]/[1 - tan A tan B] RHS = [tan A + tan B]/[1 - tan A tan B] =[(sin A/cos A) + (sin B/cos B)]/[1-(sin A/cos A)(sin B/cos B) = [sin A cos B + cos A sin B]/[cos A cos B][1 - sin A sin B/(cos A cos B)] = sin (A+B)/{[cos A cos B][cos A cos B - sin A sin B]/(cos A cos B)} = sin (A+B)/[cos A cos B - sin A sin B = sin (A+B)/cos (A+B) = tan (A+B) = LHS. Proved. Thanks. source:Quara

5 pirates of different ages have a treasure of 100 gold coins. On their ship, they decide to split the coins using this scheme: The oldest pirate proposes how to share the coins, and ALL pirates (including the oldest) vote for or against it. If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain. As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard. Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?

Algebra is great fun - you get to solve puzzles! With computer games you play by running, jumping or finding secret things. Well, with Algebra you play with letters, numbers and symbols, and you also get to find secret things! And once you learn some of the "tricks", it becomes a fun challenge to work out how to use your skills in solving each "puzzle".

Well ever wondered how the  Surface Area of a Sphere  was derived? Well here is a great visualisation to alter your perception. Step 1 : Cut the sphere in the following way. Step 2 : Spread the cut out part across the paper Step 3 : Collate the pieces together in the following way Step 4 : Spread the areas out separately to form a sine curve Step 5 : The area of the sine curve is the surface area of the sphere Here’s a  GIF file  for better understanding. There’s of course the generic method of calculation where the surface area is calculated by cutting the sphere into infinitesimally thin disks of varying radius stacked over one another and integrating them, but this above method is a different way to look at the same problem. Image Source: Google Images 220 Loves 284. Yes, this pair of numbers was considered as a symbol of love in medieval period, lovers used to send flowers, fruits to each other with these numbers written. Also so

The answer is four. I can quite confidently say that based on the wording given, the answer is four. First of all, we need to establish some things. I am assuming: All five people were alive before entering The four people that were killed are the only people that have died I can figure this out because of the last word: “remains.” “Remains” can be one of two things: a noun or a verb. As a verb , there are multiple answers to the question. First, if there are five people and four are killed, one person is left alive. However, you are in there as well, so would that be two remaining? Or should we count everyone in the room, which would be six people? Well, none of this matters. This is because the question states “How many remains?” As a verb, “remains” applies to only a few certain subjects: he/she/one (he remains/she remains/one remains). Otherwise it would be I remain/You remain/They remain/We remain. If “remain” was to be used as a verb, it would’ve said “How man

I will provide two methods for this. Method 1  (Easy way) : USE A CALCULATOR. You will get the answer in a matter of seconds like I got. Clearly, the answer is 072. Method 2  (Slight harder way) : Here, I want to find the answer without using the calculator. Let's try it. Note :- I will be making use of  Congruence Modulo  and  Euler's Theorem , so these are the prerequisites. Another way to put the question is  “Find the remainder when  2 2017 2 2017  is divided by 1000”. First of all we factorise 1000 as: 1000 = 2 3 × 5 3 1000 = 2 3 × 5 3 Next, we find the remainder by  2 3 2 3  and  5 3 5 3 seperately. It's obvious that  2 3 2 3  divides  2 2017 2 2017 . Hence, 2 2017 ≡ 0 ( m o d 8 ) 2 2017 ≡ 0 ( m o d 8 ) Now to find the remainder by 125 (or 5^3), we use Euler's Theorem. Euler's theorem is applicable in this case since  g c d ( 2 , 125 ) = 1 g c d ( 2 , 125 ) = 1 . Φ ( 125 ) = 125 ( 1 − 1 5 ) = 100 Φ ( 125 ) = 125 ( 1 − 1 5 ) = 10