3004.sagews

gcd(515,2005)

5

factor(2005)

5 * 401

[valuation(117,p) for p in prime_range(2,23)]

[0, 2, 0, 0, 0, 1, 0, 0]

next_prime(2005)

2011

d,u,v = xgcd(12,15)

[d,u,v]

[3, -1, 1]

[binomial(11,m) for m in range(12)]

[1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]

2^111 mod 7

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_4.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Ml4xMTEgbW9kIDc="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpvjv0rE/___code___.py", line 3 _sage_const_2 **_sage_const_111 mod _sage_const_7 ^ SyntaxError: invalid syntax

x, y = var('x, y')
solve([x+y==6, x-y==4], x, y)

[[x == 5, y == 1]]

divisors(28); sum(divisors(28)); 2*28

[1, 2, 4, 7, 14, 28] 56 56

n = 2005
for i in range(1000):
     n = 3*odd_part(n) + 1
       print i
       if odd_part(n)==1:
          break

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_9.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("biA9IDIwMDUKZm9yIGkgaW4gcmFuZ2UoMTAwMCk6CiAgICAgbiA9IDMqb2RkX3BhcnQobikgKyAxCiAgICAgICBwcmludCBpCiAgICAgICBpZiBvZGRfcGFydChuKT09MToKICAgICAgICAgIGJyZWFr"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpPA48Nj/___code___.py", line 6 print i ^ IndentationError: unexpected indent