Răspuns:
Explicație pas cu pas:
numerele naturale se termina in 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
patratele perfecte se termina in 0, 1, 4, 9, 6, 5
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a)
U(2^1) = 2
U(2^2) = 4
U(2^3) = 8
U(2^4) = 6
U(2^5) = 2
ultima cifra se repeta din 4 in 4
1981 : 4 = 495 rest 1
U(2^1981) = 2
nici un patrat perfect nu se termia in 2
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b)
U(3^1) = 3
U(3^2) = 9
U(3^3) = 7
U(3^4) = 1
U(3^5) = 3
ultima cifra se repeta din 4 in 4
483 : 4 = 120 rest 3
U(3^483) = 7
nici un patrat perfect nu se termina in 7
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c)
U(97^143) = U(7^143)
U(7^1) = 7
U(7^2) = 9
U(7^3) = 3
U(7^4) = 1
U(7^5) = 7
ultima cifra se repeta din 4 in 4
143 : 4 = 35 rest 3
U(97^143) = 3
nici un patrat perfect nu se termina in 3
_________________
d)
U(1998^781) = U(8^781)
U(8^1) = 8
U(8^2) = 4
U(8^3) = 2
U(8^4) = 6
U(8^5) = 8
ultima cifra se repeta din 4 in 4
781 : 4 = 195 rest 1
U(1998^781) = 8
nici un patrat perfect nu se termina in 8
________________
e)
83 : 4 = 20 rest 3
68 : 4 = 17 rest 0
U(3^83) = 7
U(8^68) = 6
U(3^83 + 8^68) = U(7 + 6) = U(13) = 3
nici un patrat perfect nu se termina in 3
