Question

Write 74057 as a product of two primes. No calculators please.

Answer 1

split the number into its prime decomposition

Answer 2

How would you do that?

Answer 3

try to divide it by the lowest prime

Answer 4

and go up

Answer 5

lol that's a lot of primes

Answer 6

It's not divisible by 2, 3, 5, 7

Answer 7

I will give a hint. Think quadratic expression.

Answer 8

im not sure, im just trying and slooking for a pattern in the results

Answer 9

By the way the hint I gave is one way to do it. There might be other ways.

Answer 10

quadratic expression?

Answer 11

i wonder what base

Answer 12

it might be in terms of 10^2

Answer 13

hmm

Answer 14

this doesn't look easy hmm
\[74057 = x^2-y^2\]

Answer 15

ooh ooh pick me !

Answer 16

\[7x^2+40x+57=(7x+19)(x+3)\]

Answer 17

Girl power!

Answer 18

yay for girls~

Answer 19

put \(x=100\) and you find that
\[74057=(700+19)(100+3)\]

Answer 20

thats very neat xD

Answer 21

I was going to say factor 72409
But I didn't like that one because it couldn't be written as a product of two primes

Answer 22

then it would be a lot harder

Answer 23

\[72409=7 \cdot 10^4+24 \cdot 10^2+9 \\ 7 \cdot 10^4+3 \cdot 10^2+21 \cdot 10^2+9 \\ 10^2(7 \cdot 10^2+3)+3(7 \cdot 10^2+3) \\ (7 \cdot 10^2+3)(10^2+3)\]
I wonder if there is a cute way to factor 703

Answer 24

(7x+3)(x+3)

Answer 25

\[703 = x^2-y^2\]
\[x^2 - 703 = y^2\]
since 27 is the least positive integer that makes the left hand side positive, we test x= 27,28,29,... that makes the left hand side a perfect square

x = 28 gives
\[28^2 - 703 = 81 = 9^2\]
so x = 28 and y = 9 will work

Answer 26

sorry was stuffing my face
that is a cute method
\[703=x^2-y^2=(28-9)(28+9)=(19)(37)\]

Answer 27

changing the base to 9 or 8 might make life a bit easy/complicated...

Answer 28

sounds scary

Answer 29

\[(703)_{10} = (861)_9\]

7x^2+3 is not factorable but we can factor 8x^2+6x+1

Answer 30

well
sqrt 703 <27 :|
not too much numbers to check

Answer 31

and also there is a thing
703 is odd
703=2n+1=(n+1)^2-1^2
703=r 4n+3
=(2n+1)^2-1^2
or n=175
(2n+3)^2-2(n+1) failed
lets try
703=8n+7--->n=87
=(8n+7)^2+(41n+16)
(41n+16)=3 mod 10 failed :|
703=10n+3 --->n=7
(10n+7)^2-(1390n+6) failed

703=12n+7 >>> n=58
(12n+7)^2-(156n+42) failed

703=14n+3
(14n+3)^2-(70n+6)
failed

703=16n+15-->n=43
(16n+15)^2-(464n+210) failed

lol too much k since 28 works :P

Answer 32

i rather to use this :-

Answer 33

sieve of eratosthenes always works but how would you know when to give up testing ?

Answer 34

when sqrt n<k such that kn+q failed