Menu Close

Category: Number Theory

Find-the-number-of-rational-numbers-r-0-lt-r-lt-1-such-that-when-r-is-written-as-fraction-in-lowest-term-The-numerator-and-demominator-have-a-sum-of-1000-

Tinku Tara 0 Comments

Question Number 110783 by Aina Samuel Temidayo last updated on 30/Aug/20 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{numbers} \\ $$$$\mathrm{r},\:\mathrm{0}<\mathrm{r}<\mathrm{1},\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{r}\:\mathrm{is}\:\mathrm{written} \\ $$$$\mathrm{as}\:\mathrm{fraction}\:\mathrm{in}\:\mathrm{lowest}\:\mathrm{term}.\:\mathrm{The} \\ $$$$\mathrm{numerator}\:\mathrm{and}\:\mathrm{demominator}\:\mathrm{have}\:\mathrm{a} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{1000}. \\ $$ Commented by mr…

Read more →

Let-a-b-and-c-be-positive-integers-such-that-ab-1-bc-1-and-bc-1-ca-1-Show-that-ab-1-is-the-sum-of-two-squares-

Tinku Tara 0 Comments

Question Number 110775 by Aina Samuel Temidayo last updated on 30/Aug/20 $$\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{ab}+\mathrm{1}\mid\mathrm{bc}+\mathrm{1}\:\mathrm{and}\:\mathrm{bc}+\mathrm{1}\mid\mathrm{ca}+\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{ab}+\mathrm{1}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{squares}. \\ $$ Terms of Service Privacy Policy…

Read more →

Let-a-b-and-c-be-positive-integers-such-that-ab-1-bc-1-and-bc-1-ca-1-Show-that-ab-1-is-the-sum-of-two-squares-

Tinku Tara 0 Comments

Question Number 110688 by Aina Samuel Temidayo last updated on 30/Aug/20 $$\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{ab}+\mathrm{1}\mid\mathrm{bc}+\mathrm{1}\:\mathrm{and}\:\mathrm{bc}+\mathrm{1}\mid\mathrm{ca}+\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{ab}+\mathrm{1}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{squares}. \\ $$ Terms of Service Privacy Policy…

Read more →

The-Diophantine-equation-x-2-y-2-1-N-xy-1-has-infinitely-many-integer-solutions-if-N-equals-Any-help-please-

Tinku Tara 0 Comments

Question Number 110644 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{The}\:\mathrm{Diophantine}\:\mathrm{equation} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{1}\:=\mathrm{N}\left(\mathrm{xy}+\mathrm{1}\right)\:\mathrm{has} \\ $$$$\mathrm{infinitely}\:\mathrm{many}\:\mathrm{integer} \\ $$$$\mathrm{solutions}\:\mathrm{if}\:\mathrm{N}\:\mathrm{equals}? \\ $$$$\mathrm{Any}\:\mathrm{help}\:\mathrm{please}? \\ $$ Commented…

Read more →

Find-the-sum-of-all-positive-two-digit-integers-that-are-divisible-by-each-of-their-digits-

Tinku Tara 0 Comments

Question Number 110591 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{positive} \\ $$$$\mathrm{two}−\mathrm{digit}\:\mathrm{integers}\:\mathrm{that}\:\mathrm{are}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{each}\:\mathrm{of}\:\mathrm{their}\:\mathrm{digits}. \\ $$ Answered by Rasheed.Sindhi last updated on 30/Aug/20…

Read more →

Let-x-and-y-be-integers-such-that-xy-1-x-2-y-and-y-2-x-i-Show-that-p-xy-1-and-p-x-2-y-then-p-y-2-x-where-p-is-a-prime-ii-Let-p-be-a-prime-Suppose-that-p-x-2-y-and-p-y-2-x-must-p-xy-1-

Tinku Tara 0 Comments

Question Number 110562 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{Let}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{be}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{xy}\neq\mathrm{1},\:\mathrm{x}^{\mathrm{2}} \neq\mathrm{y}\:\mathrm{and}\:\mathrm{y}^{\mathrm{2}} \neq\mathrm{x}. \\ $$$$ \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{p}\mid\mathrm{xy}−\mathrm{1}\:\mathrm{and}\:\mathrm{p}\mid\mathrm{x}^{\mathrm{2}} −\mathrm{y} \\ $$$$\mathrm{then}\:\mathrm{p}\mid\mathrm{y}^{\mathrm{2}} −\mathrm{x}\:\mathrm{where}\:\mathrm{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}. \\…

Read more →

Let-n-N-Using-the-formula-lcm-a-b-ab-gcd-a-b-and-lcm-a-b-c-lcm-lcm-a-b-c-find-all-the-possible-value-of-6-lcm-n-n-1-n-2-n-3-n-n-1-n-2-n-3-

Tinku Tara 0 Comments

Question Number 110565 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{Let}\:\mathrm{n}\in\mathbb{N}.\:\mathrm{Using}\:\mathrm{the}\:\mathrm{formula}\:\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right) \\ $$$$=\:\frac{\mathrm{ab}}{\mathrm{gcd}\left(\mathrm{a},\mathrm{b}\right)}\:\mathrm{and}\:\mathrm{lcm}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right) \\ $$$$=\mathrm{lcm}\left(\mathrm{lcm}\left(\mathrm{a},\mathrm{b}\right),\mathrm{c}\right),\:\mathrm{find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{possible} \\ $$$$\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{6}\bullet\mathrm{lcm}\left(\mathrm{n},\mathrm{n}+\mathrm{1},\mathrm{n}+\mathrm{2},\mathrm{n}+\mathrm{3}\right)}{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)\left(\mathrm{n}+\mathrm{3}\right)} \\ $$ Commented by kaivan.ahmadi last updated…

Read more →

Find-the-sum-of-the-nth-term-of-the-series-1-1-2-3-1-4-5-6-1-7-8-9-

Tinku Tara 0 Comments

Question Number 44986 by Tawa1 last updated on 07/Oct/18 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{nth}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series}:\:\:\:\:\frac{\mathrm{1}}{\mathrm{1}.\mathrm{2}.\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{4}.\mathrm{5}.\mathrm{6}}\:+\:\frac{\mathrm{1}}{\mathrm{7}.\mathrm{8}.\mathrm{9}}\:+\:… \\ $$ Commented by tanmay.chaudhury50@gmail.com last updated on 07/Oct/18 $${pls}\:{write}\:{the}\:{answer}\:{of}\:{questiins}\:{snd}\:{source}\:{of}\: \\ $$$${question}… \\ $$ Answered…

Read more →