A Few questions on tricks...

[Michael T]

Hey, these are just a few things I've never been able to figure out...

1) N^2 + (n+1)^2. (Ex: 165^2 + 166^2) Is there an easy way to do this?

2) Finding # of relatively prime #'s to a number, less than that number (or a different number)

3) Finding the sum of the positive integral divisors of a number

4) How you would go about finding the product of the coefficients of (4a + 7b)^2, besides the obvious 16 x 56 x 49.

5) And for approximations, what value should I be using for e? just 2.7?

Thanks,
Michael

[bradp]

1) write 1, divide multiple of 5 by 5 - take that number and multiply it by the other number and write that
35^2 + 36^2
write 1
35/5 = 7
7*36 = 252
2521
Never seen one as big as 165 and 166

2) http://math-magic.com/misc/rel_prime.htm
find the prime factorization (bases and exponents)
a. take each base, subtract 1, multiply them all together
b.take each base/exponent combo, subtract 1 frm exponent, evaluate, multiply them all together
multiply a and b together

72 = 2^3 * 3^2
2-1 = 1
3-1 = 2
2^(3-1) = 4
3^(2-1) = 3

1*2*4*3 = 24

3) http://math-magic.com/misc/pos_int_div2.htm
find prime factorization
take each part, add 1 to exponent, evaluate, subtract 1, divide by base minus 1
multiply each together

if something is to the first power, it becomes b+1 where b is the base

(b^2 - 1) / (b-1) = b+1

72
(2^(3+1) - 1) / (2-1) = 15

(3^(2+1) -1 ) / (3-1) = 13
13 * 15 = 195

4) http://forum.texasmath.org/viewtopic.php?t=9

5) at least 5 decimal places or you are slacking

[AuSmith]

1) [unparseable or potentially dangerous latex formula]. So, I like to think of it as adding one to twice the product.

5) I usually skipped the approximations if they were very icky. You should at least know these:

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

... and [unparseable or potentially dangerous latex formula]

... and [unparseable or potentially dangerous latex formula]

[Kurt]

[stupidityismygam]

less known trick is the sum of all positive relatively prime integers of n less than n

(i hope this is right i just derived it)

is [unparseable or potentially dangerous latex formula]

so for example with sum of all positive relatively prime integers of 42 less than 42

would be [unparseable or potentially dangerous latex formula]

I hope this is right

[bradp]

It is correct. some tests in 2000 or 2001 have that type of question

if you factor the 45 into 3,3, and 5 and change the 3*3*(1-(1/3)) to 3(3-1) and change the 1/5 the same way then it turns into my (math-magic's really) method :D

[stupidityismygam]