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Category: Number Theory

39-mod-4-

Question Number 15377 by Joel577 last updated on 10/Jun/17 $$−\mathrm{39}\:\mathrm{mod}\:\mathrm{4}\:=\:? \\ $$ Answered by Tinkutara last updated on 10/Jun/17 $$−\mathrm{39}\:\equiv\:−\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{4}\right) \\ $$$$\equiv\:\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{4}\right) \\ $$$$\mathrm{You}\:\mathrm{can}\:\mathrm{check}:\:−\mathrm{39}\:=\:\mathrm{4}\left(−\mathrm{10}\right)\:+\:\mathrm{1} \\…

Question-15322

Question Number 15322 by tawa tawa last updated on 09/Jun/17 Commented by mrW1 last updated on 09/Jun/17 $$\mathrm{sin}^{\mathrm{2}} \:\mathrm{x}_{\mathrm{k}} =\frac{\mathrm{4}}{\mathrm{2015}} \\ $$$$\mathrm{cos}^{\mathrm{2}} \:\mathrm{x}_{\mathrm{k}} =\mathrm{1}−\frac{\mathrm{4}}{\mathrm{2015}}=\frac{\mathrm{2011}}{\mathrm{2015}} \\…

Light-version-of-Q-13724-Expansion-of-100-has-24-0-s-at-the-end-Find-the-first-non-zero-digit-from-right-100-d000-00-What-is-the-value-of-d-

Question Number 15292 by RasheedSoomro last updated on 09/Jun/17 $$\mathrm{Light}\:\mathrm{version}\:\mathrm{of}\:\mathcal{Q}#\mathrm{13724} \\ $$$$\mathcal{E}\mathrm{xpansion}\:\mathrm{of}\:\mathrm{100}!\:\mathrm{has}\:\mathrm{24},\:\mathrm{0}'\mathrm{s}\:\:\mathrm{at}\:\mathrm{the}\:\mathrm{end}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}-\mathrm{zero}\:\mathrm{digit}\:\mathrm{from}\:\mathrm{right}. \\ $$$$\mathrm{100}!=…..\mathrm{d000}…\mathrm{00} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{d}? \\ $$ Answered by RasheedSoomro last updated…

Find-the-largest-prime-factor-of-203203-Anyone-please-suggest-the-method-without-calculators-or-log-tables-

Question Number 14949 by Tinkutara last updated on 05/Jun/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{prime}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{203203}. \\ $$$$\mathrm{Anyone}\:\mathrm{please}\:\mathrm{suggest}\:\mathrm{the}\:\mathrm{method} \\ $$$$\mathrm{without}\:\mathrm{calculators}\:\mathrm{or}\:\mathrm{log}\:\mathrm{tables}. \\ $$ Commented by RasheedSoomro last updated on 05/Jun/17 $$\mathrm{203203}=\mathrm{203000}+\mathrm{203}=\mathrm{203}\left(\mathrm{1000}+\mathrm{1}\right) \\…

Solve-1000-5-10-15-1000-x-mod-10-

Question Number 14757 by RasheedSoomro last updated on 04/Jun/17 $$\mathrm{Solve}: \\ $$$$\frac{\mathrm{1000}!}{\mathrm{5}×\mathrm{10}×\mathrm{15}×…\mathrm{1000}}\equiv\mathrm{x}\left(\mathrm{mod}\:\mathrm{10}\right) \\ $$ Answered by RasheedSoomro last updated on 05/Jun/17 $$\mathrm{x}\equiv\frac{\mathrm{1000}!}{\mathrm{5}×\mathrm{10}×\mathrm{15}×…\mathrm{1000}}\left(\mathrm{mod}\:\mathrm{10}\right. \\ $$$$\mathrm{x}\equiv\left(\mathrm{1}.\mathrm{2}.\mathrm{3}.\mathrm{4}\right)\left(\mathrm{6}.\mathrm{7}.\mathrm{8}.\mathrm{9}\right)…\left(\mathrm{996}.\mathrm{997}.\mathrm{998}.\mathrm{999}\right)\left(\mathrm{mod}\:\mathrm{10}\right) \\…

If-5-doesn-t-divide-any-of-n-n-1-n-2-n-3-then-prove-that-n-n-1-n-2-n-3-24-mod100-

Question Number 14614 by RasheedSoomro last updated on 03/Jun/17 $$\mathrm{If}\:\:\mathrm{5}\:\:\mathrm{doesn}'\mathrm{t}\:\mathrm{divide}\:\mathrm{any}\:\mathrm{of}\:\mathrm{n},\mathrm{n}+\mathrm{1}, \\ $$$$\mathrm{n}+\mathrm{2},\mathrm{n}+\mathrm{3}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)\left(\mathrm{n}+\mathrm{3}\right)\equiv\mathrm{24}\left(\mathrm{mod100}\right) \\ $$ Commented by mrW1 last updated on 03/Jun/17 $$\mathrm{n}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{divide}\:\mathrm{n}? \\…

How-many-digits-will-there-be-in-875-16-

Question Number 145359 by imjagoll last updated on 04/Jul/21 $$\mathrm{How}\:\mathrm{many}\:\mathrm{digits}\:\mathrm{will}\:\mathrm{there}\:\mathrm{be} \\ $$$$\mathrm{in}\:\mathrm{875}^{\mathrm{16}} \:? \\ $$ Answered by Olaf_Thorendsen last updated on 04/Jul/21 $$\mathrm{If}\:\mathrm{N}\:\mathrm{is}\:\mathrm{a}\:\mathrm{n}−\mathrm{digit}\:\mathrm{number}\:: \\ $$$$\mathrm{N}\:=\:{a}_{{n}−\mathrm{1}}…