Question

Determinati valoarea numarului natural x stiind ca $7^{x}$ + $7^{x + 2}$ = 2450.
Determinati ultima cifra a produsului p = $3^{71}$ × $4^{62}$
Aratati ca numarul a= $( 3^{21} + 3^{20} + 3^{19} ) : 39$ este patrat perfect.

Answer 1

1)
2450 = 49 * 50 = 7² * 50

$7^{x}+7^{x+2}= 7^{2}*50 \\ 7^{x}(1+ 7^{2} )=7^{2}*50 \\7^{x}*50=7^{2}*50 \\ 7^{x}=7^{2} \\ x=2$

2)
p = 3⁷¹ × 4⁶² = 3³⁺⁶⁸ × 4⁶² = 3³ × 3⁶⁸ × 4⁶² = 27 × (3⁴)¹⁷ × (4²)³¹ = 27 × 81¹⁷ × 16³¹  
U(27 × 81¹⁷ × 16³¹ ) = U(7 × 1 × 6) = U(42) = 2  

3)
$a= \frac{( 3^{21}+3^{20}+3^{19})}{39}= \frac{ 3^{18}* (3^{3}+3^{2}+3^{1})}{39}=\frac{ 3^{18}* (27+9+3)}{39}= \frac{ 3^{18}* (39)}{39}=3^{18} \\ 3^{18}=3^{2*9}=(3^{2})^{9} = 3^{9}*3^{9}$

cctd