Perfect numbers
Mathematicians have been fascinated for millenniums by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). Such numbers are called perfect numbers. A perfect number is a whole number, an integer greater than zero and is the sum of its proper positive devisors, that is, the sum of the positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, including itself. For example; 6, the first perfect number, its factors are 1, 2, 3 and 6. So, 1 + 2 + 3 = 6 or alternatively (1+2+3+6) divided by 2, which = 6. The next perfect number is 28, and its factors are 1, 2, 4, 7, 14, and 28. So, if we use the method which excludes the perfect number 28, we express the equation as, 1+2+4+7+14 = 28. Or using the other method, (1+2+4+7+14+28) divided by 2, it results as 28, thus making 28 a perfect number.

Throughout history, there have been studies on perfect numbers. It is not clarified when perfect numbers were first studied and it’s possible that the first studies may go back to the earliest era when numbers first aroused curiosity. It is assumed, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked. Perfect numbers were studied by Pythagoras and his followers, mainly for their mystical properties than for their number theoretic properties. Although, the four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries. The first recorded mathematical result concerning perfect numbers which is known occurs in Euclid's Elements written around 300BC. Also, it’s quite interesting to know that, not all perfect numbers were found in the same order, the ninth, tenth, and eleventh perfect numbers were found after the twelfth was discovered. The 29th was found...

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THE DIVINITY OF NUMBER:
The Importance of Number in the Philosophy of Pythagoras
by
Br. Paul Phuoc Trong Chu, SDB
Pythagoras and his followers, the Pythagoreans, were profoundly fascinated with numbers. In this paper, I will show that the heart of Pythagoras’ philosophy centers on numbers. As true to the spirit of Pythagoras, I will demonstrate this in seven ways. One, the principle of reality is mathematics and its essence...

...and theorems in a way that does not require the reader to have an in-depth knowledge of the theory of numbers but also touches upon matters of deep mathematical signiﬁcance. A companion website (www.cambridge.org/davenport) provides more details of the latest advances and sample code for important algorithms. Reviews of earlier editions: ‘. . . the well-known and charming introduction to number theory . . . can be recommended both for independent study and as a...

...About This Number?
0 is the additive identity.
1 is the multiplicative identity.
2 is the only even prime.
3 is the number of spatial dimensions we live in.
4 is the smallest number of colors sufficient to color all planar maps.
5 is the number of Platonic solids.
6 is the smallest perfectnumber.
7 is the smallest number of sides of a regular polygon that is not constructible by...

...prime. if n is odd and n>=3, then I can factorize like this:
(a + 1) (a^2 - a + 1)
and both numbers are greater than one, so a^3+1 can't be prime.
What if n is 6?
a^6 + 1
Well now I can just write that as (a^2)^3 + 1
which we know will factorize the same way:
((a^2) + 1) ( (a^2)^2 - (a^2) + 1)
The same idea will work for any number with an odd factor of 3 or more.
Therefore the only numbers n that can possibly lead to primes are those...

...REAL NUMBERS
Q.1 Determine the prime factorization of the number 556920. (1 Mark)
(Ans) 23 x 32 x 5 x 7 x 13 x 17
Explanation :
Using the Prime factorization, we have
556920 = 2 x 2 x 2 x 3 x 3 x 5 x 7 x 13 x 17 = 23 x 32 x 5 x 7 x 13 x 17
Q.2 Use Euclid’s division algorithm to find the HCF of 210 and 55. (1 Mark)
(Ans) 5
Explanation:
5 , Given integers are 210 and 55 such that 210 > 55. Applying Euclid’s division leema to 210 and...

...10th Real Numbers test paper
2011
1.
Express 140 as a product of its prime factors
2.
Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.
3.
Find the LCM and HCF of 6 and 20 by the prime factorization method.
4.
State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating
decimal.
5.
State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating
decimal.
6....

...he number theory or number systems happens to be the back bone for CAT preparation. Number systems not only form the basis of most calculations and other systems in mathematics, but also it forms a major percentage of the CAT quantitative section. The reason for that is the ability of examiner to formulate tough conceptual questions and puzzles from this section. In number systems there are hundreds of concepts and variations, along with...

... 3 is a number, numeral, and glyph. It is the natural number following 2 and preceding 4.
In mathematics
Three is approximately π when doing rapid engineering guesses or estimates. The same is true if one wants a rough-and-ready estimate of e, which is actually approximately 2.71828.
Three is the first odd prime number, and the second smallest prime. It is both the first Fermat prime and the first Mersenne prime, the only number...

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