Deck 4: Number Theory and Cryptography

Question 1

Find the prime factorization of 111111.

Answer 1

We see that 2 does not divide 111,111, but that 3 does divide 111,111, with 111,111/3 = 37,037. We see that 3 does not divide 37,037, and that 5 does not divide it either. We see that 7 does divide 37,037 with 37,037/7 = 5291. We see that 7 does not divide 5291, but 11 does divide 5291 with 5291 = 11 · 481. We find that 11 does not divide 481, but 13 does, with 481 = 13 · 37. Since 37 is prime, it follows that the prime factorization of 111,111 is 111,111 = 3 · 7 · 11 · 13 · 37.

Question 2

Answer 2

Question 3

Find the prime factorization of 45617.

Answer 3

Question 4

Prove or disprove that there are six consecutive composite integers.

Question 5

Question 6

What is the hexadecimal expansion of the

Question 7

Question 8

Prove or disprove that a positive integer congruent to 1 modulo 4 cannot have a prime factor congruent to

Question 9

The binary expansion of an integer is (110101)2. What is the base 10 expansion of this integer?

Question 10