Question

Fie A= 2^n+1 •5^n+1 +3• 2^n • 5^n+1 - 2^n+3 • 5^n

Afla valoarea lui n pentru care A are exact 50 de divizori naturali!

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Answer 1

A= 2^n+1 •5^n+1 +3• 2^n • 5^n+1 - 2^n+3 • 5^n

A = 2^n × 2^1 × 5^n × 5^1 + 3×2^n×5^n×5^1 - 2^n × 2^3 × 5^n

A = (2 ×5)^n × 2×5 + (2×5)^n × 3×5 - (2×5)^n × 8

A = 10^n × 10 + 10^n × 15 - 10^n × 8

A = 10^n × (10 + 15 - 8)

A = 10^n × 17
__________

10^n × 17 = 2^n × 5^n × 17

(n + 1) × (n + 1) × (1 + 1)= 50

(n + 1)^2 × 2 = 50

(n + 1)^2 = 50 : 2

(n + 1)^2 = 25

(n + 1)^2 = 5^2

n + 1 = 5

n = 4
_____


A = 2^4 × 5^4 × 17

(4 + 1) × (4+1) × (1 + 1) = 5 × 5 × 2 = 50 divizori