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Category: Number Theory

If-a-b-c-are-in-Harmonic-progression-find-the-value-of-a-b-b-a-b-c-b-c-

Question Number 82628 by jagoll last updated on 23/Feb/20 $${If}\:{a},{b},\:{c}\:{are}\:{in}\:{Harmonic}\:{progression} \\ $$$${find}\:{the}\:{value}\:{of}\:\frac{{a}+{b}}{{b}−{a}}\:+\:\frac{{b}+{c}}{{b}−{c}}\:.\:? \\ $$ Commented by $@ty@m123 last updated on 23/Feb/20 $$\frac{\mathrm{1}}{{a}},\:\frac{\mathrm{1}}{{b}},\frac{\mathrm{1}}{{c}}\:{are}\:{in}\:{AP} \\ $$$$\Rightarrow\frac{\mathrm{1}}{{b}}−\frac{\mathrm{1}}{{a}}=\frac{\mathrm{1}}{{c}}−\frac{\mathrm{1}}{{b}}=\boldsymbol{{d}},\:{say} \\…

Find-the-number-of-positive-integers-less-than-or-equal-to-300-that-are-multiples-of-3-or-5-but-are-not-multiples-of-10-or-15-

Question Number 16880 by Tinkutara last updated on 27/Jun/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{less}\:\mathrm{than}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{300}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{multiples}\:\mathrm{of}\:\mathrm{3}\:\mathrm{or}\:\mathrm{5},\:\mathrm{but}\:\mathrm{are}\:\mathrm{not} \\ $$$$\mathrm{multiples}\:\mathrm{of}\:\mathrm{10}\:\mathrm{or}\:\mathrm{15}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Find-out-first-non-five-digit-from-right-in-the-expansion-of-1-3-5-625-

Question Number 15889 by RasheedSoomro last updated on 15/Jun/17 $$\mathrm{Find}\:\mathrm{out}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{non}}-\boldsymbol{\mathrm{five}}\:\boldsymbol{\mathrm{digit}}\:\mathrm{from}\:\mathrm{right} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}×\mathrm{3}×\mathrm{5}×…×\mathrm{625}\right). \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Number-of-decimal-digits-in-50-is-

Question Number 15835 by RasheedSoomro last updated on 15/Jun/17 $$\:\mathrm{Number}\:\mathrm{of}\: \\ $$$$\mathrm{decimal}\:\:\mathrm{digits}\: \\ $$$$\mathrm{in}\:\:\mathrm{50}!\:\mathrm{is} \\ $$ Commented by RasheedSoomro last updated on 15/Jun/17 $$\mathrm{By}\:\mathrm{decimal}\:\mathrm{digits}\:\mathrm{I}\:\mathrm{mean}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{10}-\mathrm{base}\:\mathrm{system}. \\…

Find-all-rational-solution-of-the-equation-a-b-ab-

Question Number 15788 by tawa tawa last updated on 13/Jun/17 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{rational}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\:\mathrm{a}\:+\:\mathrm{b}\:=\:\mathrm{ab} \\ $$ Commented by mrW1 last updated on 14/Jun/17 $$\mathrm{b}=\frac{\mathrm{n}}{\mathrm{m}} \\ $$$$\mathrm{a}+\frac{\mathrm{n}}{\mathrm{m}}=\mathrm{a}×\frac{\mathrm{n}}{\mathrm{m}} \\ $$$$\left(\frac{\mathrm{n}}{\mathrm{m}}−\mathrm{1}\right)\mathrm{a}=\frac{\mathrm{n}}{\mathrm{m}}…

Prove-that-2-2-2n-1-2-2-2n-1-is-never-a-prime-for-any-positive-n-

Question Number 15789 by tawa tawa last updated on 13/Jun/17 $$\mathrm{Prove}\:\mathrm{that}:\:\:\mathrm{2}^{\mathrm{2}^{\mathrm{2n}\:+\:\mathrm{1}} \:} +\:\mathrm{2}^{\mathrm{2}^{\mathrm{2n}} } \:+\:\mathrm{1}\:,\:\:\mathrm{is}\:\mathrm{never}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{for}\:\mathrm{any}\:\mathrm{positive}\:\mathrm{n}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Find-the-four-digits-number-such-that-4-abcd-dcba-

Question Number 15787 by tawa tawa last updated on 13/Jun/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{four}\:\mathrm{digits}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{4}\:\centerdot\:\mathrm{abcd}\:=\:\mathrm{dcba} \\ $$ Commented by mrW1 last updated on 13/Jun/17 $$\mathrm{I}\:\mathrm{found}\:\mathrm{only}\:\mathrm{4}×\mathrm{2178}=\mathrm{8712} \\ $$ Commented by…

This-question-is-posted-on-the-request-of-mrW1-See-comments-of-my-answer-to-Q-15543-Find-the-last-last-non-zero-digit-of-the-expansion-of-2000-

Question Number 15742 by RasheedSoomro last updated on 14/Jun/17 $$\mathrm{This}\:\mathrm{question}\:\mathrm{is}\:\mathrm{posted}\:\mathrm{on}\:\mathrm{the}\:\mathrm{request}\:\mathrm{of}\:\mathrm{mrW1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{See}\:\mathrm{comments}\:\mathrm{of}\:\mathrm{my}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{Q}#\mathrm{15543}\right). \\ $$$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{last}\:\:\boldsymbol{\mathrm{non}}-\boldsymbol{\mathrm{zero}}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\:\mathrm{2000}! \\ $$ Commented by mrW1 last updated…