Question

CMR 3^101-7^53 chia hết cho 10

Answer 1

Sửa đề: Chứng minh (3¹⁰¹ + 7⁵³) ⋮ 10

Ta có:

3¹⁰¹ = 3.(3¹⁰)¹⁰

7⁵³ = (7⁷)⁷.7⁴

*) 3 ≡ 3 (mod 10)

3¹⁰ ≡ 9 (mod 10)

⇒ (3¹⁰)¹⁰ ≡ 9¹⁰ (mod 10) ≡ 1 (mod 10)

⇒ 3¹⁰¹ ≡ 3.(3¹⁰)¹⁰ (mod 10) ≡ 3.1 (mod 10) ≡ 3 (mod 10)

*) 7⁴ ≡ 1 (mod 10)

7⁷ ≡ 3 (mod 10)

⇒ (7⁷)⁷ ≡ 3⁷ (mod 10) ≡ 7 (mod 10)

⇒ 7⁵³ ≡ 7⁴.(7⁷)⁷ (mod 10) ≡ 1.7 (mod 10) ≡ 7 (mod 10)

⇒ 3¹⁰¹ + 7⁵³ ≡ 3 + 7 (mod 10) ≡ 10 (mod 10) ≡ 0 (mod 10)

Vậy (3¹⁰¹ + 7⁵³) ⋮ 10