What is prime factorization?
955219
ok so prime factorization is basically factoring a number into it's multiple primes. for example, to prime factorize 12, u'd figure out what it's multiples are- 4 and 3 (or 6&2..1&12 u get the point), ok now break down 4 to prime numbers- 2 and 2. and since 3 is already prime u don't have to do anything to it. so the prime factorization of 12 is 2^2 and 3
726076
Yep, it's 202.  Factor out the x and add the resulting cofactors\nin pairs whose sums are 100.\n\nx[(1+99)+(2+98)+...+\n(49+51)+50+100] =\nx[5050]  Now factor 5050 to its primes.\n\n  you get  2,5,5,101 so all you need are two factors\nof 2 and 101 to get all primes in pairs. Thus the answer is 202.\n\n202X4950=1020100 = (1010)^2\n\n Pretty clean , eh?
460136
Actually the best way to find greatest common factor is to prime factorize. \n\nThe prime factorization of 12 is:\n2^2 x 3\n\nThe prime factorization of 90 is: \n2 x 3^2 x 5\n\nThe most shared numbers in each of those prime factorizations is one 2 and one 3. \n\nMultiply these two numbers:\n\n2x3=6\n\n6 is the greatest common factor.
74381
Don't you think that maybe this so called "message board is not really for you" meaning you are not the target market of this "dumb" room...\n\nok the answer to your question is this\n\n\nfor (x=1; x<0; x = x++){\n println("the highest prime number is: " + x + " as of " + getdate());\n}\n\nthe largest prime number is x if this loop ever stop...
914600
1272 = 2 x 2 x 2 x 3 x 53   ...prime factors\n2, 3, 4, 6, 8, 12, 24, 53, 106, 159, 212, 318, 424, 636,   ...all factors\noh, I forgot, and 1 and 1272.
14699
Yes.\n\nFor any finite set of primes, there will always exist another prime not in that set.  So there can never be a "complete" finite set of all prime numbers.\n\nTake all known primes:\nP1, P2, P3,... Pn\nwhere Pn is the greatest prime.\n\n(P1 x P2 x P3 x.... x Pn) + 1 is not divisible by any of the known primes.\n\nIf it is not prime, then its prime factors are not in the original known set of primes.\n\nIf it is prime, then it itself is a prime that is not in the original known set of primes.
236680
A perfect square is the product of a whole number times itself, therefore, there is at least one whole number (its square root) which evenly factors into the number.  Since that is the case, the number can't be prime.\n\nOr you could argue the other way, a prime number is a number for which there is no other whole number (other than itself and 1) which can be a factor of it.  Since a perfect square has at least its square root as an factor, a prime cannot be a perfect square.
7923
There isn't one. Mathematicians are working on it, but they have yet to find a pattern that covers all prime numbers (except for the fact they are divisible only by themselves and one.) There are patterns that cover some prime numbers, such as paired primes (where the difference between the two prime numbers is 2, such as 5 and 7), but there is no currently known way to quickly produce or check primes, which has important implications for the RSA algorithm (used in PGP for example), which relies on the complexity of primes for its security.
696574
First figure the prime factors of each:\n88 / 2 = 44\n44 / 2 = 22\n22 / 2 = 11\n11 / 11 = 1\n\nSo 88 = 2 x 2 x 2 x 11\n\n104 / 2 = 52\n52 / 2 = 26\n26 / 2 = 13\n13 / 13 = 1\n\nSo 104 = 2 x 2 x 2 x 13\n\nThe common factors are all combinations of 2 x 2 x 2.\n\nThe common factors are therefore:\n1, 2, 4, 8 (several people forgot 1 as a common factor).\n\n(And the *greatest* common factor is 8.)
154296
Megami of Nekro is mostly correct, but I will try to clarify the situation a bit. \n\n1. factorize, or factor\n\nIf you have an algebraic expression that can be described as the product of some simpler expressions, to rewrite the expression as this product is to factorize it. \n\nSuppose you want to factorize the polynomial x^2 -3*x. In the case of polynomials of some degree, we regard polynomials of lower degree as simpler. In this case, the polynomial x^2-3*x is of degree 2, because the biggest power of x is 2 (the x^2 term). The lower degree polynomials in this case are of degree 0, called numbers, and of degree 1, called monomials. We can write the polynomial x^2-3*x as the product of lower order polynomials in the following way:\n\nx^2-3*x=x(x-3)\n\nThe two monomials x and (x-3) are factors of the polynomial x^2-3*x, and by expressing the polynomial as a product of two lower order polynomials, we have "factorized it" or "factored it". \n\n\nHere is another one:\n\nx^4+2x^2+1 is factorized as \n(x^2+1)(x^2+1)\n\nWith grade 8 mathematics, you cannot go any farther than this. We replaced a 4th degree polynomial with the product of its two factors, two 2nd degree polynomials (remember, we have decided that lower degree polynomials are simpler)\n\n\nAside: It is also interesting to note that one can factor whole numbers sometimes into a product of smaller whole numbers, which we regard as simpler. Therefore, \n\n10=5*2\n\nand 24=2*2*2*3\n\nThis is also called factoring. Numbers that can be expressed this way are called composite numbers. Numbers that cannot be expressed this way are called prime numbers (the real definition is slightly more complicated: you can look it up).\n\n2. Expand\n\nTo expand an expression is usually the opposite of factorizing. You multiply all the parts of the expression, at least as much as you can, using the distributive law, etc. \n\nIn the case of the expression x(x-3), we multiply our two monomials and return to the polynomial x^2-3*x:\n\nx(x-3)=x^2-3*x\n\nHere is another example. To expand\n\nx*(y-2)*(z+5)*y\n\nwe just multiply these factors together, \n\n x*(y-2)*(z+5)*y=\n(xy-2x)*(zy+5y)=\nxzy^2+5xy^2-2xzy-10xy\n\n\n\n3. Simplify\n\nThere is some ambiguity about what it means to simplify an expression. Computer algebra programs that simplify expressions sometimes use one definition of the simplest expression, and sometimes another. It all really depends on what shows up the features you are more interested in at the time. \n\nUsually to simplify an expression you have to do things like:\n\na. add up any like terms\n\nTo simplify  5x + 2y+ 2x+3+4, we add the terms with just numbers in them, and add the terms with just xs in them:\n\n5x + 2y+ 2x+3+4=7x+2y+7\n\nb. Multiply any numbers that can be multiplied\n\n5*2*x+2+y simplified is usually 10x+y+2\n\nc. Usually, when one simplifies an expression, the big powers are presented first, and the small powers follow (or sometimes vice versa).\n\nx+x^3+x^2 can be simplified as either \n\nx^3+x^2+x or x+x^2+x^3, depending on taste, the needs of the moment, etc.\n\n\nd. Usually when simplifying, any numbers or polynomials in ratios that can be divided will be\n\nSo 10/6 is often simplified as 5/3\n\nAnd (x^2-3*x)/x is often simplified as x-3\n\nHopefully that should get you started.
948657
Without leaving anything out, find the greatest common factor of 45, 72, 63 and 36, which is 9.  So 9 care packages.\n\nRelatively prime means their GCF is 1.  The only multiples of 3 that are divisible by 6 are even ones, and if the first number were odd it would still be odd when you multiplied it by 3, so the first number must be even.  That means the second MUST be odd because if BOTH were even, they'd habe a common factor of 2.
115942
Break each number into prime factors (factors that are prime numbers).\n12=2*2*3\n16=2*2*2*2\nThen, counting each duplicate factor once, multiply those factors together.\n2*2*3*2*2=48
773695
Look in your text book under "factoring polynomials"... or you could try this web site that explains this fairly thoroughly...\n\nhttp://faculty.ed.umuc.edu/~swalsh/Math%20Articles/Factoring.html
835669
It's almost 2/3, but not exactly.\n\nInstead 667/1000 is already in lowest terms.\n\nThe prime factors of 667 are 23 x 29\nThe prime factors of 1000 are 2 x 2 x 2 x 5 x 5 x 5.\n\nThey have none in common so you can't reduce it further.
899336
84, without even needing to check. That is 2*2*3*7, which gives 3 prime factors. You can't get 4, since 2*3*5*7 is too big.
583419
yes..http://primes.utm.edu/curios/page.php/101.html
959197
I have factored it please refer to.\n\nFrom this solution can be found