Question

Show that every positive odd integer is in the form 4q+1 4q+3

Answer 1

Answer:

Any positive odd integer is of the form 4q+1 or 4q+3. show that any positive odd integers is of the form 4 q + 1 or 4 q + 3 where q is a positive integer.

Step-by-step explanation:

Answer 2

Let be any positive integer

Let be any positive integerWe know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying, where.

Let be any positive integerWe know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying, where.Take

Let be any positive integerWe know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying, where.Take Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.

Let be any positive integerWe know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying, where.Take Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.That is, can be , where q is the quotient.

Let be any positive integerWe know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying, where.Take Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.That is, can be , where q is the quotient.Since is odd, cannot be 4q or 4q + 2 as they are both divisible by 2.

Let be any positive integerWe know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying, where.Take Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.That is, can be , where q is the quotient.Since is odd, cannot be 4q or 4q + 2 as they are both divisible by 2.Therefore, any odd integer is of the form 4q + 1 or 4q + 3.