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Number TheoryQuestion and Answers: Page 14

Question Number 101418 Answers: 0 Comments: 6

Find 2020 term from series (1/1),(2/1),(1/2),(3/1),(2/2),(1/3),(4/1),(3/2),(2/3),(1/4) ,(5/1),(4/2),... is ___ (A) ((2019)/(2020)) (B) ((61)/4) (C)((63)/1) (D) ((96)/4) (E) ((2020)/(2019))

$$\mathrm{Find}\:\mathrm{2020}\:\mathrm{term}\:\mathrm{from}\:\mathrm{series} \\ $$$$\frac{\mathrm{1}}{\mathrm{1}},\frac{\mathrm{2}}{\mathrm{1}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{3}}{\mathrm{1}},\frac{\mathrm{2}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{3}},\frac{\mathrm{4}}{\mathrm{1}},\frac{\mathrm{3}}{\mathrm{2}},\frac{\mathrm{2}}{\mathrm{3}},\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$,\frac{\mathrm{5}}{\mathrm{1}},\frac{\mathrm{4}}{\mathrm{2}},...\:\mathrm{is}\:\_\_\_ \\ $$$$\left(\mathrm{A}\right)\:\frac{\mathrm{2019}}{\mathrm{2020}}\:\:\:\:\:\:\left(\mathrm{B}\right)\:\frac{\mathrm{61}}{\mathrm{4}}\:\:\:\:\:\left(\mathrm{C}\right)\frac{\mathrm{63}}{\mathrm{1}} \\ $$$$\left(\mathrm{D}\right)\:\frac{\mathrm{96}}{\mathrm{4}}\:\:\:\:\:\:\left(\mathrm{E}\right)\:\frac{\mathrm{2020}}{\mathrm{2019}} \\ $$

Question Number 101043 Answers: 1 Comments: 0

((1+(1/2^(11) ) +(1/3^(11) ) +(1/4^(11) ) + ...)/(1−(1/2^(11) )+(1/3^(11) )−(1/4^(11) )+...)) =?

$$\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{11}} }\:+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{11}} }\:+\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{11}} }\:+\:...}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{11}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{11}} }−\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{11}} }+...}\:=? \\ $$

Question Number 100888 Answers: 0 Comments: 0

Find all pairs (k,n) of positive integer for which 7^k −3^n divides k^4 +n^2 .

$${Find}\:{all}\:{pairs}\:\left({k},{n}\right)\:{of}\:{positive}\: \\ $$$${integer}\:{for}\:{which}\:\mathrm{7}^{{k}} −\mathrm{3}^{{n}} \:{divides} \\ $$$${k}^{\mathrm{4}} +{n}^{\mathrm{2}} \:. \\ $$

Question Number 100733 Answers: 0 Comments: 6

A positive integer such as 4334 is a palindrome if it reads the same forwards or backwards. What is the only prime palindrome with an even number of digits?

$$\mathrm{A}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such}\:\mathrm{as}\:\mathrm{4334}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{palindrome}\:\mathrm{if}\:\mathrm{it}\:\mathrm{reads}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{forwards}\:\mathrm{or}\:\mathrm{backwards}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{only}\:\mathrm{prime}\:\mathrm{palindrome}\:\mathrm{with}\:\mathrm{an} \\ $$$$\mathrm{even}\:\mathrm{number}\:\mathrm{of}\:\mathrm{digits}?\: \\ $$

Question Number 100677 Answers: 0 Comments: 0

for m,n positive integers m > n prove that lcd(m,n) + lcd(m+1,n+1) > ((2mn)/(√(m−n)))

$$\mathrm{for}\:\mathrm{m},\mathrm{n}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{m}\:>\:\mathrm{n}\: \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{lcd}\left(\mathrm{m},\mathrm{n}\right)\:+\:\mathrm{lcd}\left(\mathrm{m}+\mathrm{1},\mathrm{n}+\mathrm{1}\right)\:>\:\frac{\mathrm{2mn}}{\sqrt{\mathrm{m}−\mathrm{n}}} \\ $$

Question Number 100583 Answers: 0 Comments: 0

A transformation f on a complex plane is defined by z′ = (1 +i)z −3 + 4i show that f is a simultitude with radius r and centre Ω to be determined. Determine to the invariant point under f.

$$\:\mathrm{A}\:\mathrm{transformation}\:{f}\:\mathrm{on}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{plane} \\ $$$$\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:{z}'\:=\:\left(\mathrm{1}\:+{i}\right){z}\:−\mathrm{3}\:+\:\mathrm{4}{i} \\ $$$$\:\mathrm{show}\:\mathrm{that}\:{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{simultitude}\:\mathrm{with}\:\mathrm{radius}\:{r}\:\mathrm{and}\:\mathrm{centre} \\ $$$$\Omega\:\mathrm{to}\:\mathrm{be}\:\mathrm{determined}. \\ $$$$\mathrm{Determine}\:\mathrm{to}\:\mathrm{the}\:\mathrm{invariant}\:\mathrm{point}\:\mathrm{under}\:{f}. \\ $$

Question Number 100179 Answers: 2 Comments: 0