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Category: Number Theory

ab-b-ba-a-cde-ab-b-ba-a-f-a-b-c-d-e-f-are-all-different-and-in-some-order-consecutive-also-Determine-the-remain

Question Number 200315 by Rasheed.Sindhi last updated on 17/Nov/23 $$\:\begin{cases}{\overline {\:{ab}\:}\centerdot\overline {\:{b}\:}+\overline {\:{ba}\:}\centerdot\overline {\:{a}\:}=\overline {\:{cde}\:}}\\{\overline {\:{ab}\:}\centerdot\overline {\:{b}\:}−\overline {\:{ba}\:}\centerdot\overline {\:{a}\:}=\overline {\:{f}\:}\:}\end{cases} \\ $$$${a},{b},{c},{d},{e},{f}\:{are}\:{all}\:{different}\:{and}\:{in} \\ $$$${some}\:{order}\:{consecutive}\:{also}. \\…

By-strong-induction-prove-that-any-natural-number-equal-to-or-bigger-than-8-can-be-written-as-3a-5b-where-a-and-b-are-non-negative-integers-

Question Number 200041 by depressiveshrek last updated on 12/Nov/23 $${By}\:{strong}\:{induction}\:{prove}\:{that}\:{any} \\ $$$${natural}\:{number}\:{equal}\:{to}\:{or}\:{bigger}\:{than} \\ $$$$\mathrm{8}\:{can}\:{be}\:{written}\:{as}\:\mathrm{3}{a}+\mathrm{5}{b}\:{where}\:{a}\:{and}\:{b} \\ $$$${are}\:{non}−{negative}\:{integers}. \\ $$ Answered by des_ last updated on 12/Nov/23…

Find-the-number-of-positive-integers-that-are-factors-of-3-19-7-12-10-25-and-are-also-multiples-of-3-15-7-10-10-19-

Question Number 199311 by necx122 last updated on 01/Nov/23 $${Find}\:{the}\:{number}\:{of}\:{positive}\:{integers} \\ $$$${that}\:{are}\:{factors}\:{of}\:\mathrm{3}^{\mathrm{19}} .\mathrm{7}^{\mathrm{12}} .\mathrm{10}^{\mathrm{25}} \:{and}\:{are} \\ $$$${also}\:{multiples}\:{of}\:\mathrm{3}^{\mathrm{15}} .\mathrm{7}^{\mathrm{10}} .\mathrm{10}^{\mathrm{19}} \\ $$ Answered by AST last…

Sum-of-two-irrational-numbers-is-1-less-than-their-product-and-8-less-than-their-sum-of-squares-Find-the-larger-of-the-two-numbers-

Question Number 199011 by necx122 last updated on 26/Oct/23 $${Sum}\:{of}\:{two}\:{irrational}\:{numbers}\:{is}\:\mathrm{1} \\ $$$${less}\:{than}\:{their}\:{product},\:{and}\:\mathrm{8}\:{less}\:{than} \\ $$$${their}\:{sum}\:{of}\:{squares}.\:{Find}\:{the}\:{larger} \\ $$$${of}\:{the}\:{two}\:{numbers}. \\ $$ Commented by nikif99 last updated on 27/Oct/23…

20-11-1-mod-1000-

Question Number 198400 by cortano12 last updated on 19/Oct/23 $$\:\:\mathrm{20}^{\mathrm{11}} −\mathrm{1}\:=\:…\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$ Answered by MM42 last updated on 19/Oct/23 $$\mathrm{20}^{\mathrm{11}} −\mathrm{1}\overset{\mathrm{1000}} {\equiv}−\mathrm{1}\overset{\mathrm{1000}} {\equiv}\mathrm{999} \\…