# How To Prove It – Exercise 0.1

Oct 02 2012

Solutions to Exercises in the Introduction of How To Prove It by Daniel J Velleman.

Problem (1.a): Factor $2^{15}-1 = 32767$ into a product of  two smaller positive integers.

Solution:

Here the number to be factored is of the form $2^n-1$

So here $n = 15 = 3 \times 5$

Let $a = 3, b = 5$

So here:

$x = 2^b-1$

$= 2^5-1 = 31$

$y = 1 + 2^b + 2^{2b} + 2^{3b} + . . . + 2^{(a-1)b}$

$= 1 + 2^5 + 2^{2 \times 5}$

$= 1057$

$\therefore 32767 = 31 \times 1057$

Problem (1.b): Find an integer $x$ such that $1 < x < 2^{32767} - 1$ and $2^{32767}$ is divisible by $x$

Solution:

From the previous result:

$32767 = 31 \times 1057$

$\therefore x = 2^{31} - 1 = 2147483647$

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