Contact Style About

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

By Kaylee

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

About the author
Kaylee

02. Read the following passage carefully and answer the questions that follow
Of man’s carliest inventions we know very littl

evaluate the following products without multiplying directly 105×96​

2 thoughts on “Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.”

  2. Step-by-step explanation:

    Sol:– Let a be a positive integer q be the quotient and r be the remainder

    Dividing a by 3 using the Euclid’s Division Lemma

    we have

    a = 3q + r,

    where a = ≤ r < 3

    Putting r = 0, 1 and 2, we get:

    a = 3q

    → a^2 = 9q^2

    = 3 × 3q^2

    = 3m (Assuming m = q^2)

    Then, a = 3q + 1

    a^2 = (3q + 1)^2

    = 9q^2 + 6q + 1

    = 3(3q^2 + 2q) + 1

    = 3m +1 (Assuming m = 3q^2 + 2q)

    Next, a = 3q + 2

    → a² = (3q + 2)^2

    = 9q^2 + 12q + 4

    = 9q^2 + 12q +3+1

    = 3(3q^2 + 4q + 1) + 1

    = 3m + 1. (Assuming m = 3q^2 + 4q+1)

    Therefore, the square of any positive integer (say a^2) is always of the form 3m or 3m + 1

    Hence, proved

    Reply

Leave a Comment