Question and Answers Forum

All Questions      Topic List

Vector Questions

Previous in All Question      Next in All Question      

Previous in Vector      Next in Vector      

Question Number 27651 by Joel578 last updated on 12/Jan/18

A positive number has 8 distinct divisors Lets say a, b, c, d, e, f, g and h Given a . b . c . d . e . f . g . h = 3111696 Find that number

$$\mathrm{A}\:\mathrm{positive}\:\mathrm{number}\:\mathrm{has}\:\mathrm{8}\:\mathrm{distinct}\:\mathrm{divisors} \\ $$$$\mathrm{Lets}\:\mathrm{say}\:{a},\:{b},\:{c},\:{d},\:{e},\:{f},\:{g}\:\mathrm{and}\:{h} \\ $$$$\mathrm{Given}\:\:{a}\:.\:{b}\:.\:{c}\:.\:{d}\:.\:{e}\:.\:{f}\:.\:{g}\:.\:{h}\:=\:\mathrm{3111696} \\ $$$$\mathrm{Find}\:\mathrm{that}\:\mathrm{number} \\ $$

Commented by Joel578 last updated on 12/Jan/18

What is distinct divisor? Can we say that is it same as factor?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{distinct}\:\mathrm{divisor}?\:\mathrm{Can}\:\mathrm{we}\:\mathrm{say}\:\mathrm{that}\:\mathrm{is} \\ $$$$\mathrm{it}\:\mathrm{same}\:\mathrm{as}\:\mathrm{factor}? \\ $$

Commented by Rasheed.Sindhi last updated on 14/Jan/18

Distinct divisors means differnt divisors and yes divisor is same as factor. If a∣c & b∣c & a≠b then a and b are distinct divisors.

$$\mathrm{Distinct}\:\mathrm{divisors}\:\mathrm{means}\:\mathrm{differnt} \\ $$$$\mathrm{divisors}\:\mathrm{and}\:\mathrm{yes}\:\mathrm{divisor}\:\mathrm{is}\:\mathrm{same} \\ $$$$\mathrm{as}\:\mathrm{factor}. \\ $$$$\mathrm{If}\:\:\mathrm{a}\mid\mathrm{c}\:\&\:\mathrm{b}\mid\mathrm{c}\:\&\:\mathrm{a}\neq\mathrm{b}\:\mathrm{then}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b} \\ $$$$\mathrm{are}\:\boldsymbol{\mathrm{distinct}}\:\boldsymbol{\mathrm{divisors}}. \\ $$$$ \\ $$

Answered by mrW2 last updated on 14/Jan/18

The number is 2×3×7=42. It has exactly 8 (distinct) divisors: 1 (=a) 2 (=b) 3 (=c) 6 (=d) 7 (=e) 14 (=f) 21 (=g) 42 (=h) The product of all those divisors is a∙b∙c∙d∙e∙f∙g∙h=1×2×3×6×7×14×21×42=311696

$${The}\:{number}\:{is}\:\mathrm{2}×\mathrm{3}×\mathrm{7}=\mathrm{42}.\:{It}\:{has} \\ $$$${exactly}\:\mathrm{8}\:\left({distinct}\right)\:{divisors}: \\ $$$$\mathrm{1}\:\left(={a}\right) \\ $$$$\mathrm{2}\:\left(={b}\right) \\ $$$$\mathrm{3}\:\left(={c}\right) \\ $$$$\mathrm{6}\:\left(={d}\right) \\ $$$$\mathrm{7}\:\left(={e}\right) \\ $$$$\mathrm{14}\:\left(={f}\right) \\ $$$$\mathrm{21}\:\left(={g}\right) \\ $$$$\mathrm{42}\:\left(={h}\right) \\ $$$${The}\:{product}\:{of}\:{all}\:{those}\:{divisors}\:{is} \\ $$$${a}\centerdot{b}\centerdot{c}\centerdot{d}\centerdot{e}\centerdot{f}\centerdot{g}\centerdot{h}=\mathrm{1}×\mathrm{2}×\mathrm{3}×\mathrm{6}×\mathrm{7}×\mathrm{14}×\mathrm{21}×\mathrm{42}=\mathrm{311696} \\ $$

Commented by Rasheed.Sindhi last updated on 13/Jan/18

Hi Sir mrW1∣ mrW2 Your answer is perfect as it contains all the divisors of required number. I deleted my answer because its logic was deffective.

$$\mathrm{Hi}\:\mathrm{Sir}\:\mathrm{mrW1}\mid\:\mathrm{mrW2} \\ $$$$\mathrm{Your}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{perfect}\:\mathrm{as}\:\mathrm{it}\:\mathrm{contains} \\ $$$$\mathrm{all}\:\mathrm{the}\:\mathrm{divisors}\:\mathrm{of}\:\mathrm{required}\:\mathrm{number}. \\ $$$$\mathrm{I}\:\mathrm{deleted}\:\mathrm{my}\:\mathrm{answer}\:\mathrm{because}\:\mathrm{its} \\ $$$$\mathrm{logic}\:\mathrm{was}\:\mathrm{deffective}. \\ $$

Commented by mrW2 last updated on 13/Jan/18

Sir, to be honest, I didn′t catch the question really till I saw your working. It brought me finally to the right understanding and to the answer. Therefore thank you for your working!

$${Sir},\:{to}\:{be}\:{honest},\:{I}\:{didn}'{t}\:{catch}\:{the} \\ $$$${question}\:{really}\:{till}\:{I}\:{saw}\:{your}\:{working}. \\ $$$${It}\:{brought}\:{me}\:{finally}\:{to}\:{the}\:{right}\: \\ $$$${understanding}\:{and}\:{to}\:{the}\:{answer}.\: \\ $$$${Therefore}\:{thank}\:{you}\:{for}\:{your}\:{working}! \\ $$

Commented by Rasheed.Sindhi last updated on 13/Jan/18

ThAnKs SiR! Could You Help Me in solving Q#27422 & Q27627.

$$\mathbb{T}{h}\mathbb{A}{n}\mathbb{K}{s}\:\mathbb{S}{i}\mathbb{R}!\: \\ $$$$\:\:\:\mathbb{C}\mathrm{ould}\:\mathbb{Y}\mathrm{ou}\:\mathbb{H}\mathrm{elp}\:\mathbb{M}\mathrm{e}\:\mathrm{in}\:\mathrm{solving} \\ $$$$\mathbb{Q}#\mathrm{27422}\:\&\:\mathbb{Q}\mathrm{27627}. \\ $$$$ \\ $$

Answered by mrW2 last updated on 14/Jan/18

An other way to solve this question: We know the number, let′s say N has 8 different divisors a, b, c..., h. We know if number k is a divisor of N, then (N/k) must be an other divisor of it, the product of both divisors is N. This is to say if we arrange a, b,c...,h increasingly, we′ll have a∙h=N b∙g=N c∙f=N d∙e=N Then we′ll get a∙b∙c∙d∙e∙f∙g∙h=N^4 =3111696 (as given) according to Sir Rasheed.Sindhi: 3111696=(2×3×7)^4 ⇒N=2×3×7=42

$${An}\:{other}\:{way}\:{to}\:{solve}\:{this}\:{question}: \\ $$$${We}\:{know}\:{the}\:{number},\:{let}'{s}\:{say}\:{N}\:{has} \\ $$$$\mathrm{8}\:{different}\:{divisors}\:{a},\:{b},\:{c}...,\:{h}. \\ $$$$ \\ $$$${We}\:{know}\:{if}\:{number}\:{k}\:{is}\:{a}\:{divisor}\:{of} \\ $$$${N},\:{then}\:\frac{{N}}{{k}}\:{must}\:{be}\:{an}\:{other}\:{divisor}\:{of} \\ $$$${it},\:{the}\:{product}\:{of}\:{both}\:{divisors}\:{is}\:{N}. \\ $$$$ \\ $$$${This}\:{is}\:{to}\:{say}\:{if}\:{we}\:{arrange}\:{a},\:{b},{c}...,{h} \\ $$$${increasingly},\:{we}'{ll}\:{have} \\ $$$${a}\centerdot{h}={N} \\ $$$${b}\centerdot{g}={N} \\ $$$${c}\centerdot{f}={N} \\ $$$${d}\centerdot{e}={N} \\ $$$${Then}\:{we}'{ll}\:{get}\: \\ $$$${a}\centerdot{b}\centerdot{c}\centerdot{d}\centerdot{e}\centerdot{f}\centerdot{g}\centerdot{h}={N}^{\mathrm{4}} =\mathrm{3111696}\:\left({as}\:{given}\right) \\ $$$${according}\:{to}\:{Sir}\:{Rasheed}.{Sindhi}:\:\mathrm{3111696}=\left(\mathrm{2}×\mathrm{3}×\mathrm{7}\right)^{\mathrm{4}} \\ $$$$\Rightarrow{N}=\mathrm{2}×\mathrm{3}×\mathrm{7}=\mathrm{42} \\ $$

Commented by Rasheed.Sindhi last updated on 14/Jan/18

Creative work Sir! This is realy a ′ method ′for such problems!

$$\boldsymbol{{Creative}}\:\boldsymbol{{work}}\:\boldsymbol{{Sir}}! \\ $$$$\:{This}\:{is}\:{realy}\:{a}\:'\:\boldsymbol{{method}}\:'{for}\: \\ $$$${such}\:{problems}!\: \\ $$

Commented by Joel578 last updated on 14/Jan/18

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com