Răspuns :
[tex]\displaystyle\bf\\0^2=0\\1^2=1\\2^2=4\\3^2=9\\4^2=16\implies~U=6\\5^2=25\implies~U=5\\6^2=36\implies~U=6\\7^2=49\implies~U=9\\8^2=62\implies~U=4\\9^2=81\implies~U=1\\\\\implies~\text{Ultima cifra a unui patrat perfect }~\in\{0;~1;~4;~5;~6;~9\}\\\\\implies~\text{NU poate fi patrat perfect un numat cu ultima cifra }\in\{2;~3;~7;~8\}[/tex]
[tex]\displaystyle\bf\\\text{Calculam ultima cifra a numarului }~x=3^{42}+3^{80}+3^{63}\\\\U(x)=U\left[U\Big(3^{42}\Big)+U\Big(3^{80}\Big)+U\Big(3^{63}\Big)\right]=\\\\\\=U\left[U\Big(3^{40+2}\Big)+U\Big(3^{80}\Big)+U\Big(3^{60+3}\Big)\right]=\\\\\\=U\left[U\Big(3^{40}\times3^2\Big)+U\Big(3^{80}\Big)+U\Big(3^{60}\times3^3\Big)\right]=\\\\\\=U\left[U\Big(3^{4\times10}\times3^2\Big)+U\Big(3^{4\times20}\Big)+U\Big(3^{4\times15}\times3^3\Big)\right]=[/tex]
[tex]\displaystyle\bf\\=U\left[U\Big(\Big(3^4\Big)^{10}\times3^2\Big)+U\Big(\Big(3^4\Big)^{20}\Big)+U\Big(\Big(3^4\Big)^{15}\times3^3\Big)\right]=\\\\\\=U\left[U\Big(81^{10}\times9\Big)+U\Big(81^{20}\Big)+U\Big(81^{15}\times27\Big)\right]=\\\\\\=U\left[U(1\times9)+1+U(1\times27)\right]=\\\\=U\left[9+1+7\right]=U(17)=\boxed{\bf7}\\\\U(x)=7~~\implies~x\neq pp[/tex]
Răspuns:
Explicație pas cu pas:
x = 3^42 + 3^80 + 3^63
3^1 se termina in 3
3^2 se termina in 9
3^3 se termina in 7
3^4 se termina in 1
3^5 se termina in 3
deci ultima cifra se repeta din 4 in 4
42 : 4 = 10 rest 2
80 : 4 = 10 rest 0
63 : 4 = 15 rest 3
3^42 se termina in 9
3^80 se termina in 1
3^63 se termina in 7
x se termina in 9 + 1 + 7 = 17, adica in 7
numar terminat in 1, la patrat se termina in 1
numar terminat in 2, la patrat se termina in 4
numar terminat in 3, la patrat se termina in 9
numar terminat in 4, la patrat se termina in 6
numar terminat in 5, la patrat se termina in 5
numar terminat in 6, la patrat se termina in 6
numar terminat in 7, la patrat se termina in 9
numar terminat in 8, la patrat se termina in 4
numar terminat in 9, la patrat se termina in 1
numar terminat in 0, la patrat se termina in 0
Nici un numar natural la patrat nu se termina in 7, deci x nu este patrat perfect.
