For you math geeks out there...

[travis2]

Here's an interesting problem for you...

Find the value of x such that the following is true:

123 (base x) = 18 (base 10)

[MannyIsGod]

Yeah, I'm going to pass on this.

Math = not my forte

[bigzak25]

in my youth...i'm out of practice....

well, i just refreshed my memory, and i'll guess x=-5 or x=3.

123 base x = 18

translates to

1*x^2 + 2*x^1 + 3*x^0=18

x^2+2x+3=18

x^2+2x-15=0

(x+5)(x-3)=0

x=-5, 3

if i'm wrong...nevermind....

[travis2]

bigzak is very close...but he's not done yet.

[bigzak25]

what do you me to do? box my answer?

seriously. gimme a nudge teach. :smokin
-----------------------------------------------------------



here i have one too...from mensa....


how many cubic yards of dirt are in a hole that is 3 ft x 3 ft x 6ft? :smokin

[travis2]

what do you me to do? box my answer?

seriously. gimme a nudge teach. :smokin
-----------------------------------------------------------



here i have one too...from mensa....


how many cubic yards of dirt are in a hole that is 3 ft x 3 ft x 6ft? :smokin

Nudge: There's only one answer.

Dirt: None. There's no dirt in a hole. :smokin

[baseline bum]

It's 3, because how are you going to have a negative base, since base to any even power is positive? It would leave lots of gaps in the integers.

[travis2]

It's 3, because how are you going to have a negative base, since base to any even power is positive? It would leave lots of gaps in the integers.

The answer is not 3.

The number 123 can't exist in base 3 (base 3 has no digit 3).

Therefore, the answer must be -5.

Using the normal rules of base conversion, it can be shown to be true:

1*(-5)^2 + 2*(-5) + 3 = 25 - 10 + 3 = 18.

Actually, I believed as you did, that a negative base was meaningless. A little research corrected my misconceptions.

One side effect of a negative base is that what we call negative numbers are in fact represented by positive numbers. You keep track by noting the number of digits...an odd number of digits and the number is positive; even number of digits, negative.

In fact, any number can be used as a base. It doesn't even have to be rational.

[baseline bum]

Shit.. my bad. My brain doesn't function correctly before 11 I think. :(

[desflood]

Now I recall why I hate math so much.

[travis2]

Shit.. my bad. My brain doesn't function correctly before 11 I think. :(

I've been told the same thing. Unfortunately I get to work about 6.

[travis2]

Now I recall why I hate math so much.

Well, if you count in base (pi), you really can say that math is an irrational subject...:lol

[GoldToe]

Thank God for computers.

[Kori Ellis]

http://www.mailbling.com/nerdalert.jpg

[bigzak25]

Dirt: None. There's no dirt in a hole.


travis, you fuck. you could have let some of the non-nerds take a guess.... :p :nerd

[travis2]

http://www.mailbling.com/nerdalert.jpg

You been talking to Jenn or something? :cry

[travis2]

travis, you fuck. you could have let some of the non-nerds take a guess.... :p :nerd

I didn't know the question wasn't for me...you were talking to me, and so I just assumed...

[bigzak25]

it's all good trav...i'm just talkin shit...good job! :smokin

[baseline bum]

OK... what is the remainder of 14 to the 763265 power when divided by 37?

I.e., 14^763265 mod 37

[travis2]

OK... what is the remainder of 14 to the 763265 power when divided by 37?

I.e., 14^763265 mod 37

May I assume base 10?

[baseline bum]

Yeah... show your work too :spin

[baseline bum]

First, factor 763,265 in binary:

2^20 = (2^10)^2 = 1024^2 = (1000+24)(1000+24) = 1,000,000 + 2 * 24,000 + 24^2
= 1,048,000 + 24^2 = 1,048,000 + (20+4)(20+4) = 1,048,000 + (400+ 2 *4 * 20 + 16)
= 1,048,000 + (400 + 160 + 16) = 1,048,576 > 763,265

2^10 = 1024 < 763,265

2^15 = 2^10 * 2^5 = 1024 * 32 = (1000+24)*(30+2) = 30,000 + 2000 + 720 + 48
= 32,768 < 763,265

2^17 = 2^15 * 2^2 = 32,768 * 4 = 32,768 * (2+2) = 65,536 + 65,536 = 131,072 < 763,265

2^18 = 2^17*2 = 262,144 < 763,265

2^19 = 2^18 * 2 = 524,288 < 763,265

Therefore, 2^19 is the largest power of 2 that fits into 763,265

763,265 - 2^19 = 763,265 - 524,288 = 238,977

Ie, 763,265 = 2^19 + 238,977

By our previous work, the power we're looking for is greater than or equal to 17 and less than 18 since 2^18 > 238,977 and 2^17 < 238,977

therefore, 2^17 is the largest power of 2 that fits into 238,977

238,977 - 2^17 = 238,977 - 131,072 = 107,905

Ie, 763,265 = 2^19 + 2^17 + 107,905

By our work in doing the binary search for 19, the biggest power of 2 that can fit into 107,905 is less than 2^17 (since 2^17 > 107,905), and greater than or equal to 2^15 since 2^15 < 107,905

2^16 = 65,536 < 107,905, so 2^16 is the largest power of 2 that can git in 107,905

107,905 - 2^16 = 107,905 - 65,536 = 42,369

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 42,369

By the same technique, we see 2^15 is the largest power of 2 that fits into 42,369 is 2^15

42,369-2^15 = 42,369 - 32,768 = 9,601

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 9,601

2^15 = 32,768 > 9,601 and 2^10 = 1,024 < 9,601

2^12 = 2^10 * 2^2 = 2^10 * 4 = (1025 -1 ) * 4 = 4100 - 4 = 4,096 < 9,601

2^13 = 2^12 * 2 = 8,192 < 9,601

Therefore, 2^13 is the largest power of 2 that fits into 9,601

9,601 - 2^13 = 9,601 - 8,192 = 1,409

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 1,409

Do another binary search... 2^10 = 1,024 < 1,409, 2^12 = 8,192 > 1,409

2^11 = 2,048 > 1,409

Therefore, 2^10 is the largest power of 2 that fits into 1,409

1,409 - 2^10 = 1,409 - 1,024 = 385

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 385

Do a binary search for the biggest power of 2 that fits into 385:

2^10 = 1,024 > 385, 2^5 = 32 < 385 ==> exponent is >= 5, less than 10

2^7 = 4 * 2^5 = 128 < 385
2^8 = 2*128 = 256 < 385
2^9 = 512 > 385

Therefore, 2^8 is the largest power of 2 that fits into 385

385 - 2^8 = 385 - 256 = 129

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 2^8 + 129

129 = 128 + 1 = 2^7 + 1 = 2^7 + 2^0

Therfore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 2^8 + 2^7 + 1
763,265 = 2^16 * (2^3 + 2 + 1) + 2^8 *( 2^7 + 2^5 + 2^2 + 1) + 2^4 * 2^3 + 1

-----------------------------------------------------------------------------------

14^763265 mod 37 = 14^(2^16 * (2^3 + 2 + 1) + 2^8 *( 2^7 + 2^5 + 2^2 + 1) + 2^4 * 2^3 + 1) mod 37

= 14^(2^16 * (2^3 + 2 + 1)) * 14^(2^8 *( 2^7 + 2^5 + 2^2 + 1)) * 14^(2^7) * 14 mod 37

14 ^ 2 mod 37 = (10+4)*(10+4) mod 37 = (100 + 80 + 16) mod 37 = 100 mod 37 + 80 mod 37 + 16 mod 37 = 26 mod 37 + 6 mod 37 + 16 mod 37 = ( 32 + 16 ) mod 37 = 11

14^4 mod 37 = 11^2 mod 37 = 121 mod 37 = 10

14^8 mod 37 = 10^2 mod 37 = 100 mod 37 = 26

14^16 mod 37 = 26^2 mod 37 = ((20+6)*(20+6)) mod 37
= (400 + 240 + 36) mod 37 = 400 mod 37 + 240 mod 37 + 36 mod 37
= 30 mod 37 + 18 mod 37 + 36 mod 37
= (30 + 18 + 36) mod 37 = (30 + 17) mod 37 = 10

14^32 mod 37 = 10^2 mod 37 = 26

14^64 mod 37 = 26^2 mod 37 = 10

14^128 mod 37 = 10^2 mod 37 = 26

ie, 14 ^ (2^k) mod 37 = {10 if k even; 26 if k odd}

Since 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 2^8 + 2^7 + 1,

14^763,265 = 14 ^ (2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 2^8 + 2^7 + 1)
= 14^(2^19) * 14^(2^17) * 14^(2^16) * 14 ^(2^15) * 14^(2^13) * 14^(2^10) * 14^(2^8) * 14^(2^7) * 14

Therefore, 14^763,265 mod 37 = 26 * 26 * 10 * 26 * 26 * 10 * 10 * 26 * 14 mod 37

26 * 26 * 10 * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (26 * 26) * 10 * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (10) * 10 * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (10 * 10) * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (26) * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (26* 26) * 26 * 10 * 10 * 26 * 14 mod 37
= (10) * 26 * (10 * 10) * 26 * 14 mod 37
= (10) * 26 * (26) * 26 * 14 mod 37
= (10) * (26*26) * 26 * 14 mod 37
= (10) * (10) * 26 * 14 mod 37
= (10*10) * 26 * 14 mod 37
= (26) * 26 * 14 mod 37
= (26*26) * 14 mod 37
= (10) * 14 mod 37
= (10*14) mod 37
= 140 mod 37

Therefore, 14^763265 mod 37 = 140 mod 37 = (37*3 + 29) mod 37 = 29

ie, the remainder of 14^763265 / 37 is 29

[scott]

there just aren't enough math sluts out there these days.

[baseline bum]

All the women I talk to who are impressed by math skills are conservative asians who don't fuck on the first date. :(

[Spurminator]

Yeah, but depending on how you work out a deal with a hooker, you might be able to get her to pay you...

[travis2]

First, factor 763,265 in binary:

2^20 = (2^10)^2 = 1024^2 = (1000+24)(1000+24) = 1,000,000 + 2 * 24,000 + 24^2
= 1,048,000 + 24^2 = 1,048,000 + (20+4)(20+4) = 1,048,000 + (400+ 2 *4 * 20 + 16)
= 1,048,000 + (400 + 160 + 16) = 1,048,576 > 763,265

2^10 = 1024 < 763,265

2^15 = 2^10 * 2^5 = 1024 * 32 = (1000+24)*(30+2) = 30,000 + 2000 + 720 + 48
= 32,768 < 763,265

2^17 = 2^15 * 2^2 = 32,768 * 4 = 32,768 * (2+2) = 65,536 + 65,536 = 131,072 < 763,265

2^18 = 2^17*2 = 262,144 < 763,265

2^19 = 2^18 * 2 = 524,288 < 763,265

Therefore, 2^19 is the largest power of 2 that fits into 763,265

763,265 - 2^19 = 763,265 - 524,288 = 238,977

Ie, 763,265 = 2^19 + 238,977

By our previous work, the power we're looking for is greater than or equal to 17 and less than 18 since 2^18 > 238,977 and 2^17 < 238,977

therefore, 2^17 is the largest power of 2 that fits into 238,977

238,977 - 2^17 = 238,977 - 131,072 = 107,905

Ie, 763,265 = 2^19 + 2^17 + 107,905

By our work in doing the binary search for 19, the biggest power of 2 that can fit into 107,905 is less than 2^17 (since 2^17 > 107,905), and greater than or equal to 2^15 since 2^15 < 107,905

2^16 = 65,536 < 107,905, so 2^16 is the largest power of 2 that can git in 107,905

107,905 - 2^16 = 107,905 - 65,536 = 42,369

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 42,369

By the same technique, we see 2^15 is the largest power of 2 that fits into 42,369 is 2^15

42,369-2^15 = 42,369 - 32,768 = 9,601

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 9,601

2^15 = 32,768 > 9,601 and 2^10 = 1,024 < 9,601

2^12 = 2^10 * 2^2 = 2^10 * 4 = (1025 -1 ) * 4 = 4100 - 4 = 4,096 < 9,601

2^13 = 2^12 * 2 = 8,192 < 9,601

Therefore, 2^13 is the largest power of 2 that fits into 9,601

9,601 - 2^13 = 9,601 - 8,192 = 1,409

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 1,409

Do another binary search... 2^10 = 1,024 < 1,409, 2^12 = 8,192 > 1,409

2^11 = 2,048 > 1,409

Therefore, 2^10 is the largest power of 2 that fits into 1,409

1,409 - 2^10 = 1,409 - 1,024 = 385

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 385

Do a binary search for the biggest power of 2 that fits into 385:

2^10 = 1,024 > 385, 2^5 = 32 < 385 ==> exponent is >= 5, less than 10

2^7 = 4 * 2^5 = 128 < 385
2^8 = 2*128 = 256 < 385
2^9 = 512 > 385

Therefore, 2^8 is the largest power of 2 that fits into 385

385 - 2^8 = 385 - 256 = 129

Therefore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 2^8 + 129

129 = 128 + 1 = 2^7 + 1 = 2^7 + 2^0

Therfore, 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 2^8 + 2^7 + 1
763,265 = 2^16 * (2^3 + 2 + 1) + 2^8 *( 2^7 + 2^5 + 2^2 + 1) + 2^4 * 2^3 + 1

-----------------------------------------------------------------------------------

14^763265 mod 37 = 14^(2^16 * (2^3 + 2 + 1) + 2^8 *( 2^7 + 2^5 + 2^2 + 1) + 2^4 * 2^3 + 1) mod 37

= 14^(2^16 * (2^3 + 2 + 1)) * 14^(2^8 *( 2^7 + 2^5 + 2^2 + 1)) * 14^(2^7) * 14 mod 37

14 ^ 2 mod 37 = (10+4)*(10+4) mod 37 = (100 + 80 + 16) mod 37 = 100 mod 37 + 80 mod 37 + 16 mod 37 = 26 mod 37 + 6 mod 37 + 16 mod 37 = ( 32 + 16 ) mod 37 = 11

14^4 mod 37 = 11^2 mod 37 = 121 mod 37 = 10

14^8 mod 37 = 10^2 mod 37 = 100 mod 37 = 26

14^16 mod 37 = 26^2 mod 37 = ((20+6)*(20+6)) mod 37
= (400 + 240 + 36) mod 37 = 400 mod 37 + 240 mod 37 + 36 mod 37
= 30 mod 37 + 18 mod 37 + 36 mod 37
= (30 + 18 + 36) mod 37 = (30 + 17) mod 37 = 10

14^32 mod 37 = 10^2 mod 37 = 26

14^64 mod 37 = 26^2 mod 37 = 10

14^128 mod 37 = 10^2 mod 37 = 26

ie, 14 ^ (2^k) mod 37 = {10 if k even; 26 if k odd}

Since 763,265 = 2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 2^8 + 2^7 + 1,

14^763,265 = 14 ^ (2^19 + 2^17 + 2^16 + 2^15 + 2^13 + 2^10 + 2^8 + 2^7 + 1)
= 14^(2^19) * 14^(2^17) * 14^(2^16) * 14 ^(2^15) * 14^(2^13) * 14^(2^10) * 14^(2^8) * 14^(2^7) * 14

Therefore, 14^763,265 mod 37 = 26 * 26 * 10 * 26 * 26 * 10 * 10 * 26 * 14 mod 37

26 * 26 * 10 * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (26 * 26) * 10 * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (10) * 10 * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (10 * 10) * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (26) * 26 * 26 * 10 * 10 * 26 * 14 mod 37
= (26* 26) * 26 * 10 * 10 * 26 * 14 mod 37
= (10) * 26 * (10 * 10) * 26 * 14 mod 37
= (10) * 26 * (26) * 26 * 14 mod 37
= (10) * (26*26) * 26 * 14 mod 37
= (10) * (10) * 26 * 14 mod 37
= (10*10) * 26 * 14 mod 37
= (26) * 26 * 14 mod 37
= (26*26) * 14 mod 37
= (10) * 14 mod 37
= (10*14) mod 37
= 140 mod 37

Therefore, 14^763265 mod 37 = 140 mod 37 = (37*3 + 29) mod 37 = 29

ie, the remainder of 14^763265 / 37 is 29

Hmmmmm...I worked it out using residues and got 8 (which is -29)...

763265=1+128+256+1024+8192+32768+65536+131072+5242 88

so 14^763265 = 14^(1+128+256+1024+8192+32768+65536+131072+524288)

14 mod 37 = -14
14^128 mod 37 = -11
14^256 mod 37 = 10
14^1024 mod 37 = 10
14^8192 mod 37 = -11
14^32768 mod 37 = -11
14^65536 mod 37 = 10
14^131072 mod 37 = -11
14^524288 mod 37 = -11

Therefore 14^763265 mod 37 = (-14)(-11)(10)(10)(-11)(-11)(10)(-11)(-11) mod 37 = 2254714000 mod 37 = 8

[spurster]

14^763265 mod 37

14^x mod 37 cycles every 12 values of x, e.g.,

14^0 mod 37 = 1
14^1 mod 37 = 14
14^2 mod 37 = 11
14^3 mod 37 = 6
14^4 mod 37 = 10
14^5 mod 37 = 29
14^6 mod 37 = 36
14^7 mod 37 = 23
14^8 mod 37 = 26
14^9 mod 37 = 31
14^10 mod 37 = 27
14^11 mod 37 = 8
14^12 mod 37 = 1 (back to 1 again)

Noting that 763265 mod 12 = 5, it follows

14^763265 mod 37 = 14^5 mod 37 = 29

[baseline bum]

Travis, you messed up on the first part... 14 mod 37 is 14, not -14... substitute 14 for -14 or -23 for -14 and mod the product by 37 and you get 29

[scott]

Duh!!!

[travis2]

My first term may be in error. I took 14 mod 37 as -14...probably should be +14.

Flipping that first sign gives the answer as -8 rather than 8. And -8 and 29 are equivalent (mod 37)

BTW, spurster, an easy cycle occurs if you expand the exponent as a binary sum, as baseline and I did. As the exponents increase (2, 4, 8, 16, 32, 64...) the residues are (11, 10, -11, 10, -11, 10...)

[travis2]

Travis, you messed up on the first part... 14 mod 37 is 14, not -14... substitute 14 for -14 or -23 for -14 and mod the product by 37 and you get 29

Yeah, I was typing as you were typing...you finished first...:lol

[Spurminator]

That's impressive and all, but can you combine a binary equation with an elephant joke?

[travis2]

That's impressive and all, but can you combine a binary equation with an elephant joke?

Probably not and keep it clean...:oops

:lol

[scott]

You guys should just splurge for Excel.

[baseline bum]

Yeah right... the opportunity cost of Excel is 212 gropes at the tip rail.

[scott]

:lol

[cherylsteele]

This is good and all....but can you balance my bank account?


You show that stuff to the teller and they would just look at you like a puzzled puppy or kitten......then still say you are $25.00 in the hole.