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Category: Number Theory

Two-different-two-digit-natural-numbers-are-written-beside-each-other-such-that-the-larger-number-is-written-on-the-left-When-the-absolute-difference-of-the-two-numbers-is-subtracted-from-the-four-di

Question Number 113005 by Aina Samuel Temidayo last updated on 10/Sep/20 $$\mathrm{Two}\:\mathrm{different}\:\mathrm{two}−\mathrm{digit}\:\mathrm{natural} \\ $$$$\mathrm{numbers}\:\mathrm{are}\:\mathrm{written}\:\mathrm{beside}\:\mathrm{each} \\ $$$$\mathrm{other}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{larger}\:\mathrm{number} \\ $$$$\mathrm{is}\:\mathrm{written}\:\mathrm{on}\:\mathrm{the}\:\mathrm{left}.\:\mathrm{When}\:\mathrm{the} \\ $$$$\mathrm{absolute}\:\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{numbers}\:\mathrm{is}\:\mathrm{subtracted}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{four}−\mathrm{digit}\:\mathrm{number}\:\mathrm{so}\:\mathrm{formed},\:\mathrm{the} \\…

The-first-23-natural-numbers-are-written-in-an-increasing-order-beside-each-other-to-form-a-single-number-What-is-the-remainder-when-this-number-is-divided-by-18-

Question Number 113000 by Aina Samuel Temidayo last updated on 10/Sep/20 $$\mathrm{The}\:\mathrm{first}\:\mathrm{23}\:\mathrm{natural}\:\mathrm{numbers} \\ $$$$\mathrm{are}\:\mathrm{written}\:\mathrm{in}\:\mathrm{an}\:\mathrm{increasing}\:\mathrm{order} \\ $$$$\mathrm{beside}\:\mathrm{each}\:\mathrm{other}\:\mathrm{to}\:\mathrm{form}\:\mathrm{a}\:\mathrm{single} \\ $$$$\mathrm{number}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when} \\ $$$$\mathrm{this}\:\mathrm{number}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{18}? \\ $$ Answered by floor(10²Eta[1])…

The-digits-of-a-three-digit-number-A-are-written-in-the-reverse-order-to-form-another-three-digit-number-B-If-B-gt-A-and-B-A-is-perfectly-divisible-by-7-Find-the-range-of-values-of-A-

Question Number 113003 by Aina Samuel Temidayo last updated on 10/Sep/20 $$\mathrm{The}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{a}\:\mathrm{three}−\mathrm{digit}\:\mathrm{number}\:\mathrm{A} \\ $$$$\mathrm{are}\:\mathrm{written}\:\mathrm{in}\:\mathrm{the}\:\mathrm{reverse}\:\mathrm{order}\:\mathrm{to} \\ $$$$\mathrm{form}\:\mathrm{another}\:\mathrm{three}−\mathrm{digit}\:\mathrm{number}\:\mathrm{B}. \\ $$$$\mathrm{If}\:\mathrm{B}>\mathrm{A}\:\mathrm{and}\:\mathrm{B}−\mathrm{A}\:\mathrm{is}\:\mathrm{perfectly} \\ $$$$\mathrm{divisible}\:\mathrm{by}\:\mathrm{7}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of} \\ $$$$\mathrm{values}\:\mathrm{of}\:\mathrm{A}. \\ $$ Commented…

N-is-completely-divisible-by-13-52-What-is-the-sum-of-the-digits-of-the-smallest-such-number-N-

Question Number 113001 by Aina Samuel Temidayo last updated on 10/Sep/20 $$\mathrm{N}!\:\mathrm{is}\:\mathrm{completely}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{13}^{\mathrm{52}} . \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{smallest}\:\mathrm{such}\:\mathrm{number}\:\mathrm{N}? \\ $$ Commented by MJS_new last updated on…

If-x-y-z-1-and-x-y-z-are-positive-real-numbers-then-the-least-value-of-1-x-1-1-y-1-1-z-1-is-

Question Number 112998 by Aina Samuel Temidayo last updated on 10/Sep/20 $$\mathrm{If}\:\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{1}\:\mathrm{and}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\mathrm{are}\:\mathrm{positive} \\ $$$$\mathrm{real}\:\mathrm{numbers},\:\mathrm{then}\:\mathrm{the}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of} \\ $$$$\left(\frac{\mathrm{1}}{\mathrm{x}}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{\mathrm{y}}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{\mathrm{z}}−\mathrm{1}\right)\:\mathrm{is}\: \\ $$ Answered by MJS_new last updated on 11/Sep/20…

Question-47310

Question Number 47310 by Meritguide1234 last updated on 08/Nov/18 Commented by MJS last updated on 08/Nov/18 $${x}\neq{y}\neq{z}\neq\mathrm{0}\:\mathrm{I}\:\mathrm{guess}. \\ $$$$\mathrm{but}\:\mathrm{we}\:\mathrm{know}\:\mathrm{that}\:\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{triple}\:{xyz}\:\mathrm{for}\:{n}>\mathrm{2} \\ $$$$\mathrm{so}\:\mathrm{we}\:\mathrm{only}\:\mathrm{have}\:\mathrm{to}\:\mathrm{show}\:\mathrm{for}\:{n}=\mathrm{2}\:\mathrm{and}\:\mathrm{in}\:\mathrm{this} \\ $$$$\mathrm{case}\:\mathrm{it}'\mathrm{s}\:\mathrm{obviously}\:\mathrm{true}\: \\ $$…