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Category: Number Theory

How-many-digits-will-there-be-in-875-16-

Question Number 145359 by imjagoll last updated on 04/Jul/21 $$\mathrm{How}\:\mathrm{many}\:\mathrm{digits}\:\mathrm{will}\:\mathrm{there}\:\mathrm{be} \\ $$$$\mathrm{in}\:\mathrm{875}^{\mathrm{16}} \:? \\ $$ Answered by Olaf_Thorendsen last updated on 04/Jul/21 $$\mathrm{If}\:\mathrm{N}\:\mathrm{is}\:\mathrm{a}\:\mathrm{n}−\mathrm{digit}\:\mathrm{number}\:: \\ $$$$\mathrm{N}\:=\:{a}_{{n}−\mathrm{1}}…

Question-continuing-from-mrW1-post-on-p-2-mod-n-1-Find-a-number-n-such-that-for-all-m-lt-n-such-that-HCF-m-n-1-m-2-mod-n-1-e-g-for-12-possible-value-of-m-are-1-5-7-11-

Question Number 13733 by prakash jain last updated on 22/May/17 $$\mathrm{Question}\:\mathrm{continuing}\:\mathrm{from} \\ $$$$\mathrm{mrW1}\:\mathrm{post}\:\mathrm{on}\:{p}^{\mathrm{2}} \:\mathrm{mod}\:\mathrm{n}\equiv\mathrm{1}. \\ $$$$\mathrm{Find}\:\mathrm{a}\:\mathrm{number}\:{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{for}\:\mathrm{all}\:{m}<{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{HCF}\left({m},{n}\right)=\mathrm{1} \\ $$$${m}^{\mathrm{2}} \:\mathrm{mod}\:{n}\:=\mathrm{1} \\ $$$${e}.{g}.\:\mathrm{for}\:\mathrm{12}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{m} \\ $$$$\mathrm{are}\:\mathrm{1},\mathrm{5},\mathrm{7},\mathrm{11}.…

1-7-142857-1-7-is-a-recurring-decimal-of-period-6-What-will-be-the-period-of-1-7-20-

Question Number 13725 by prakash jain last updated on 22/May/17 $$\frac{\mathrm{1}}{\mathrm{7}}=.\overline {\mathrm{142857}} \\ $$$$\frac{\mathrm{1}}{\mathrm{7}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{recurring}\:\mathrm{decimal}\:\mathrm{of}\:\mathrm{period}\:\mathrm{6}. \\ $$$$ \\ $$$$\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{period}\:\mathrm{of}\:\frac{\mathrm{1}}{\mathrm{7}^{\mathrm{20}} }? \\ $$ Commented by RasheedSoomro last…

Expansion-of-1000-has-249-0-s-at-the-end-Find-the-first-non-zero-digit-from-right-1000-d000-00-What-is-the-value-of-d-

Question Number 13724 by prakash jain last updated on 22/May/17 $$\mathrm{Expansion}\:\mathrm{of}\:\mathrm{1000}!\:\mathrm{has}\:\mathrm{249},\:\mathrm{0}'{s}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}−\mathrm{zero}\:\mathrm{digit}\:\mathrm{from} \\ $$$$\mathrm{right}. \\ $$$$\mathrm{1000}!=……{d}\mathrm{000}…\mathrm{00} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{d}? \\ $$ Commented by prakash jain…

Show-that-19-93-13-99-is-a-positive-integer-divisible-by-162-

Question Number 13606 by Tinkutara last updated on 21/May/17 $$\mathrm{Show}\:\mathrm{that}\:\mathrm{19}^{\mathrm{93}} \:−\:\mathrm{13}^{\mathrm{99}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{positive} \\ $$$$\mathrm{integer}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{162}. \\ $$ Answered by mrW1 last updated on 21/May/17 $$\mathrm{log}\:\left(\mathrm{19}^{\mathrm{93}} \right)=\mathrm{93}×\mathrm{log}\:\mathrm{19}\approx\mathrm{119}…

Calculate-19-93-mod-81-

Question Number 13584 by Tinkutara last updated on 21/May/17 $$\mathrm{Calculate}\:\mathrm{19}^{\mathrm{93}} \:\left(\mathrm{mod}\:\mathrm{81}\right). \\ $$ Commented by RasheedSindhi last updated on 22/May/17 $$\mathrm{Using} \\ $$$$\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{n}} \equiv\mathrm{1}+\mathrm{nx}\left(\mathrm{mod}\:\mathrm{y}\right) \\…

Let-p-be-a-prime-number-gt-3-What-is-the-remainder-when-p-2-is-divided-by-12-

Question Number 13529 by Tinkutara last updated on 20/May/17 $$\mathrm{Let}\:{p}\:\mathrm{be}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:>\:\mathrm{3}.\:\mathrm{What} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:{p}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\mathrm{12}? \\ $$ Commented by prakash jain last updated on 20/May/17…