Question

Find the principal square root of 251001 without a calculator.

Answer 1

501

Answer 2

Did you get that without a calculator?

Answer 3

Oh okay...
I made up the question purposely so if would be factorable like a quadratic in terms of the base 10^2.
I thought it would be more fun but I guess not :p


\[ 251001 \\ 25 \cdot 10^4+10 \cdot 10^2 +1 \\ (5 \cdot 10^2+1)^2 \\ (500+1)^2 \\ (501)^2 \]

Answer 4

Oh cool. I did not see that.

Answer 5

Yea I like it... :)

Like factor 156 in a similar way
1*10^2+5*10+6
(10+3)(10+2)
(13)(12)

yes I know this doesn't work for all numbers
but it is fun for the numbers it works for

Answer 6

I wrote 156 as a quadratic expression in terms of 10

Answer 7

Numbers in the form abc
where a, b, and c are digits of the number

abc can be factored like this if we can find nonnegative integers that multiplied to be a*c and also add up to be b


So we know numbers like
156
143
176
275
and so on can be factored in this way

Answer 8

so factoring 300090006 should be doable without a calculator...

Answer 9

what are the factors of that

Answer 10

\[300090006 \\ 3 00000000+90000+6 \\ 3 \cdot 10^8+9 \cdot 10^4+6 \\ 3 \cdot 10^8+3 \cdot 10^4+6 \cdot 10^4+6 \\ \text{ factor by grouping } \\ 3 \cdot 10^4(10^4+1)+6(10^4+1) \\ (3 \cdot 10^4+6)(10^4+1)\]

Answer 11

\[(30006)(10001)\]

Answer 12

that can be factored more

Answer 13

but it has been factored :p
just not completely

Answer 14

\[3(10002)(10001) \\ 3(3)(334)(10001)\]

Answer 15

Okay. And by factoring this way we can find if it's 'square root-able' , quickly.

Answer 16

oops not enough 3's in 334
3(3)(3334)(10001)

Answer 17

clearly this number is not a perfect square, since $$300090006 = (3 \cdot 10^4+6)(10^4+1)$$

Answer 18

Nevermind, that is wrong :)

Answer 19

hmmm I womder if we can say that...
like
let n be a positive integer
\[(a \cdot 10^n+b)(c \cdot 10^n+e)\]
I wonder if this is ever a perfect square besides when having the following case:
a=c and b=e

Answer 20

I honestly don't know enough about number theory to know this

Answer 21

A prime number will have no factoring.

Answer 22

I know ganeishie and some others and me talked about this a long time ago
on openstudy
but i can't remember if we talked about that

Answer 23

Quickly factor this number:
100003

Answer 24

using your method

Answer 25

\[100003 \\ 10 \cdot 10^4+3 \\ \text{ hmmm } .... 1 \cdot 10^5+3 \]
not looking good

Answer 26

but maybe i should think about thinking in terms of something other than 10

Answer 27

are you familiar with the 'brute force' way to factor a number.

Algorithm:
start with the smallest prime number 2, divide number by it,
then keep dividing 2 until it is no longer divisible by 2.

Next prime is 3, keep dividing number by 3...

Answer 28

251001
clearly not divisible by 2
next we try 3.

Answer 29

ok back

Answer 30

the answer is yes

Answer 31

251001
clearly not divisible by 2

next we try 3.
oh it divides by 3
251001 /3 = 83667

divide the quotient by 3
83667 / 3 = 27889

Answer 32

did you know @moreso that you can tell if a number is divisible by 3 by adding up the digits of that number and seeing if that sum is divisible by 3
*if yes then the original number is divisible by 3
* if no then the original number is not divisible by 3

Answer 33

251001

does it divide by 2
clearly not divisible by 2 since it has an odd number in the units place.
Next we try 3.

Is it divisible by 3?
yes it divides by 3

251001 /3 = 83667

can we divide the quotient by 3 ?
yes we can.

83667 / 3 = 27889

27889 /3 = 9296.3333 , no good.

now we move on to 5,
but the number ends in a 9. so don't check.

move on to 7

Answer 34

Right . There are many tricks.

Answer 35

yes and tricks are fun to explore

Answer 36

With a calculator a number is divisible when the decimal place is 0, a crude test.

Answer 37

were you an OS user?

Answer 38

This method is like a sieve. it is picking up all the factors.

Answer 39

I used os a few times. Yes.

Answer 40

ah.. what was your username?

Answer 41

benjc9

Answer 42

I stopped using it a year ago.

Answer 43

well dang yep i don't remember you :(

Answer 44

have you studied any algorithms in number theory
i think there are a few there

Answer 45

i think i remember something about pollard's something

Answer 46

I was sneaky, i gave you a prime number to factor.

Answer 47

lol yep i used wolfram after my quadratic trick was not working

Answer 48

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwjKzoX6wvDRAhVMzIMKHdPQB8EQFgglMAE&url=https%3A%2F%2Fmath.berkeley.edu%2F~sagrawal%2Fsu14_math55%2Fnotes_pollard.pdf&usg=AFQjCNGTp7hBszL-SPRRnVCiNYPxpzcrqQ&sig2=BUiPC9Lm2Ud-y-FtjjkOCw
yes this is a factoring algorithm

Answer 49

i can't remember it well

Answer 50

i remember they used to give lots of money for factoring really really really crazy large numbers

Answer 51

they did this i think to keep improving the security of like technology stuff

Answer 52

Oh but why does factoring make it more secure?
wouldnt it make it unsecure

Answer 53

if you are able to factor it then it is not secure
so when someone is able to factor i think they up the amount of digits to make it more secure or something like that

Answer 54

https://en.wikipedia.org/wiki/RSA_Factoring_Challenge

Answer 55

I see, but they aren't factoring by hand. They are using computers. Essentially these math people are programming algorithms.

Answer 56

read those first 2 sentences there

Answer 57

yep but they are creating factoring algorithms that computers can use to factor the numbers

Answer 58

oh didn't read your whole sentence

Answer 59

Right.

Answer 60

now i can probably type up some algorithm for my factoring method above
but my factoring method has a big chance of failing

Answer 61

for a lot of numbers

Answer 62

i would definitely not win lol

Answer 63

if you can show how your quadratic method covers all possible cases of factoring then you have created , indirectly,
a new primality test

Answer 64

Because if a number cannot be factored quadratically, it has no factorization. Thus it is prime.

Answer 65

i believe not for sure that there are some numbers not in a quadratic form but perhaps some other polynomial form that can be factored maybe

Answer 66

Also it seems to work better for larger number.

For small numbers it is hard to factor quadratically
24 = 4 * 6 = 2^2 * 6

Answer 67

like
11011

Answer 68

i haven't really looked for a quadratic form here
one sec

Answer 69

but i was thinking of
11*100+11
11*10^2+11
nevermind about that one

Answer 70

because it does have a quadratic form

Answer 71

that is factorable

Answer 72

i was thinking about other polynomials like x^4+x^3+x+1
when I wrote 11011

Answer 73

but yep it has form 11*x^2+11 which is 11(x^2+1)

Answer 74

anyways it is my bed time
i have work tomorrow

Answer 75

thanks moreso for chatting with me
i was bored

Answer 76

- thats a really old way to find square roots that works for all numbers

Answer 77

first you partition the number from the left into pairs
then you find the nearest square root less than the first pair
in this case its exactly 5 so you place 5 in the answer line (top line) and 5 to the left of vertical line
5*5 = 25 the subtract
next double th e number in answer line and place it left of vertical line
next bring down the next 2 digits (10) as shown

now we want to add 1 digit to the 10 so that 10x <= 0010
in this case it is 0 so we add 0 to the answer line
Now bring down the next 2 digits 00 so we have 001001
Now double the 50 on answer line and place it to the left
to get 1001 we need to add 1 digit to the 100 so 1 is placed on answer line
and we are finished we have the exact square root.

Answer 78

its long wined I know!!
much easier with a calculator!

Answer 79

* long winded

Answer 80

That's interesting.

Answer 81

it kinda looks like division

Answer 82

I've tried to see if there's an algebraic basis for it - if we can reconcile it with the binomial
(a + b^2 = a^2 + 2ab + b^2??

Answer 83

What @celticcat presented is something more efficient, algorithmic, and convenient as an alternative to the calculator. Thanks myininaya for posting and thanks celticcat for your solution. I figured out how to do it based on your solution.

Answer 84

For example, I figured out how to find the Square root of 66049 using Square Root Long Division Method:

Answer 85

501

Answer 86

I got 501 What i did was divided it by two and then there is you answer

Answer 87

501

Answer 88

501

Answer 89

oh wow 3 years ago

Answer 90

501

Answer 91

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Answer 92

The square root of -251001 is the number, which multiplied by itself 2 times, is -251001. In other words, this number to the power of 2 equals -251001

Answer 93

Ain't this is the wrong section