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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