Combinatorics | Number Theory Answers

Subtract the given numbers using the indicated complements. Also show your answers in non-complement form, using (+) and (-) to indicate the sign of the number.

a. 848₁₀ – 692₁₀ in 10’s complement

b. 618₁₆ – 8E3₁₆ in 16’s complement

c. 10001100₂ – 11100011₂ in 2’s complement

d. 11000101₂ – 10001110₂ in 1’s complement

e. 4020₈ – 2021₈ in 8’s complement 

Work out the following and give answers in the indicated bases/codes. (Show your ALL working):

a. 1660₈ – 530₁₆ in base 8

b. AB.75₁₆ – 830.5₁₀ in base 2

c. 4028₁₀ + 475.08₁₆ in base 16

d. 1024₁₀ – 11010111₂ in base 2 

Subtract the following numbers in the indicated bases (do not make any conversion to a different base):

a. 4607₈ - 1705₈

b. F32₁₆ - A65₁₆ 

c. 110010₂ - 010101₂

d. 1234₅ - 432₅ 

1. Add the following numbers in the indicated bases (do not make any conversion to a different base):

a. 4607.54₈ + 1705.63₈

b. A65₁₆ + F32₁₆

c. 110010₂ + 010101₂

d. 432₅ + 1234₅

How many integers form 1 to 500 are divisible by 2, 3, 4, and5?

For how many three digit numbers (100 to 999) is the sum of the digits even? (For example, 343 has an even sum of digits: 3 + 4 + 3 10 which is even.) Find the answer and explain why it is correct in at least two different ways.

Prove: Use R - 3L for divisibility by 31.

Prove: Use R + 7L to determine if 23| N.

Prove: Methods for divisibility by 13.

1. Method 2. ( R + 4L) 

Prove: 12|N iff 4| N ^ 3 | N.