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Category: Number Theory

PRIME-BIRTHDAYS-Do-you-know-Prime1611-No-no-it-s-not-an-ID-of-the-forum-member-It-is-a-person-who-was-born-on-November-16-0001-On-his-birthday-astrologers-formed-a-string-from-his-

Question Number 159461 by Rasheed.Sindhi last updated on 17/Nov/21 $$\:\:\:\:\:\:\:\:\:\:\prec\mathbb{PRIME}-\mathcal{BIRTHDAYS}\succ \\ $$$$\mathrm{Do}\:\mathrm{you}\:\mathrm{know}\:'\mathrm{Prime1611}'?… \\ $$$$\mathrm{No},\mathrm{no}\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{an}\:\mathrm{ID}\:\mathrm{of}\:\mathrm{the}\:\mathrm{forum}- \\ $$$$\mathrm{member}.\mathrm{It}\:\mathrm{is}\:\mathrm{a}\:\mathrm{person}\:\mathrm{who}\:\mathrm{was}\:\mathrm{born} \\ $$$$\mathrm{on}\:\mathrm{November}\:\mathrm{16},\:\mathrm{0001}.\mathrm{On}\:\mathrm{his} \\ $$$$\mathrm{birthday}\:\mathrm{astrologers}\:\mathrm{formed}\:\mathrm{a}\:\mathrm{string} \\ $$$$\mathrm{from}\:\mathrm{his}\:\mathrm{birthdate}\:\mathrm{in}\:\mathrm{the}\:\mathrm{way}: \\ $$$$'\mathrm{ddmmyyyy}':\:'\mathrm{16110001}'\:\mathrm{The}\:\mathrm{astro}- \\…

Find-the-number-of-positive-integers-x-such-that-x-m-1-x-m-1-for-a-particular-integer-m-2-means-G-I-F-

Question Number 28186 by Tinkutara last updated on 25/Jan/18 $${Find}\:{the}\:{number}\:{of}\:{positive}\:{integers} \\ $$$${x}\:{such}\:{that}\:\left[\frac{{x}}{{m}−\mathrm{1}}\right]=\left[\frac{{x}}{{m}+\mathrm{1}}\right],\:{for}\:{a} \\ $$$${particular}\:{integer}\:{m}\geqslant\mathrm{2}. \\ $$$$\left[\:\right]\:{means}\:{G}.{I}.{F}. \\ $$ Commented by abdo imad last updated on…

Question-27936

Question Number 27936 by Tinkutara last updated on 17/Jan/18 Commented by Rasheed.Sindhi last updated on 19/Jan/18 $$\mathrm{p}^{\mathrm{2}} +\mathrm{7pq}+\mathrm{q}^{\mathrm{2}} =\mathrm{n}^{\mathrm{2}} \:\:;\:\mathrm{n}\in\mathbb{Z} \\ $$$$\mathrm{p}^{\mathrm{2}} +\mathrm{7pq}+\mathrm{q}^{\mathrm{2}} −\mathrm{n}^{\mathrm{2}} =\mathrm{0}…

a-b-amp-c-are-distinct-primes-and-x-y-z-0-1-2-What-is-the-number-of-divisors-common-to-the-numbers-a-x-b-y-c-z-a-x-b-z-c-y-a-y-b-x-c-z-a-y-b-z-c-x-a-z-b-x-c-y-amp-a-z-b-y-c-z-

Question Number 27922 by Rasheed.Sindhi last updated on 17/Jan/18 $$\mathrm{a},\mathrm{b}\:\&\:\mathrm{c}\:\mathrm{are}\:\boldsymbol{\mathrm{distinct}}\:\boldsymbol{\mathrm{primes}}\:\mathrm{and} \\ $$$$\mathrm{x},\mathrm{y},\mathrm{z}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},…\right\}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\boldsymbol{\mathrm{divisors}}, \\ $$$$\boldsymbol{\mathrm{common}}\:\mathrm{to}\:\mathrm{the}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{y}}} \boldsymbol{\mathrm{c}}^{\boldsymbol{\mathrm{z}}} , \\ $$$$\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{z}}} \boldsymbol{\mathrm{c}}^{\boldsymbol{\mathrm{y}}} ,\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{y}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{x}}}…

a-and-b-are-distinct-primes-and-x-y-0-1-2-What-is-the-number-of-divisors-common-to-the-numbers-a-x-b-y-and-a-y-b-x-

Question Number 27888 by Rasheed.Sindhi last updated on 16/Jan/18 $$\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\boldsymbol{\mathrm{distinct}}\:\boldsymbol{\mathrm{primes}}\:\mathrm{and} \\ $$$$\mathrm{x},\mathrm{y}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},…\right\}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{divisors}} \\ $$$$\boldsymbol{\mathrm{common}}\:\mathrm{to}\:\mathrm{the}\:\boldsymbol{\mathrm{numbers}}\:\left(\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{y}}} \right) \\ $$$$\boldsymbol{\mathrm{and}}\:\left(\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{y}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{x}}} \right)? \\ $$ Commented…

If-N-is-perfect-nth-power-prove-that-n-d-N-1-Where-d-N-denotes-number-of-divisors-of-N-Also-show-by-an-example-that-its-vice-versa-is-not-necessarily-correct-

Question Number 27816 by Rasheed.Sindhi last updated on 15/Jan/18 $$\mathrm{If}\:\mathrm{N}\:\mathrm{is}\:\boldsymbol{\mathrm{perfect}}\:\boldsymbol{\mathrm{nth}}\:\boldsymbol{\mathrm{power}},\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\mathrm{n}\:\mid\:\left(\mathrm{d}\left(\mathrm{N}\right)−\mathrm{1}\right)\: \\ $$$$\left[{Where}\:\mathrm{d}\left(\mathrm{N}\right)\:{denotes}\:\boldsymbol{{number}}\right. \\ $$$$\left.\boldsymbol{{of}}\:\boldsymbol{{divisors}}\:\boldsymbol{{of}}\:\mathrm{N}\right] \\ $$$$\mathrm{Also}\:\mathrm{show}\:\mathrm{by}\:\mathrm{an}\:\mathrm{example}\:\mathrm{that}\:\mathrm{its} \\ $$$$\mathrm{vice}\:\mathrm{versa}\:\mathrm{is}\:\mathrm{not}\:\mathrm{necessarily}\:\mathrm{correct}. \\ $$ Commented by Rasheed.Sindhi…

If-the-number-of-divisors-of-a-number-is-odd-prove-that-the-number-is-perfect-square-and-vice-versa-

Question Number 27767 by Rasheed.Sindhi last updated on 14/Jan/18 $$\mathrm{If}\:\mathrm{the}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{divisors}}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{number}\:\mathrm{is}\:\boldsymbol{\mathrm{odd}},\mathrm{prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{is}\:\boldsymbol{\mathrm{perfect}}\:\boldsymbol{\mathrm{square}}\:\mathrm{and} \\ $$$$\mathrm{vice}\:\mathrm{versa}. \\ $$ Answered by mrW2 last updated on 14/Jan/18…

1-1-x-x-1-1-1-2019-2019-Find-all-possible-values-of-x-

Question Number 93241 by Ayod 19 last updated on 12/May/20 $$\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{x}+\mathrm{1}} =\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2019}}\right)^{\mathrm{2019}} \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:{x} \\ $$ Answered by prakash jain last updated on 12/May/20 $${x}=−\mathrm{2020}…