# Prove natural number

• April 25th 2009, 08:12 AM
usagi_killer
Prove natural number
Show that $\frac{9^y + 6 * 2^y}{21}$ is a natural number for all y, where y is a natural number. (Without using mathematical induction)
• April 25th 2009, 08:31 AM
red_dog
Clearly, $9^y+6\cdot 2^y \ \vdots \ 3$

$9^y+6\cdot 2^y=9^y-2^y+7\cdot 2^y=$

$=(9-2)(9^{y-1}+9^{y-2}\cdot 2+\ldots+9\cdot 2^{y-2}+2^{y-1})+7\cdot 2^y=7a+7b=7c \ \vdots \ 7$

Then the number is divisible by 21.
• April 25th 2009, 08:35 AM
usagi_killer
Thanks but what's the a, b and c for?
• April 25th 2009, 09:55 AM
stapel
Quote:

Originally Posted by usagi_killer
Thanks but what's the a, b and c for?

What is the value of 9 - 2? Note that "9 - 2" is multiplied on some other factor, the exact value (and even expression) of which we are unclear. Since this value is unknown, we can compress the (very long) expression with "a".

Note that the other term is "7 * 2^y", where "y" is unknown, and thus "2^y" is unknown. This expression may be replaced with "b".

Then the sum is 7a + 7b = 7(a + b). Replace "a + b" with "c".