a-number-consists-of-digits-1-and-2-the-sum-of-its-digits-is-2018-if-the-number-is-multiplied-with-5-the-sum-of-the-digits-will-be-10000-find-how-many-digits-this-number-has-

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Question Number 71534 by mr W last updated on 16/Oct/19

$$anumberconsistsofdigits1and2.$$$$thesumofitsdigitsis2018.$$$$ifthenumberismultipliedwith5,$$$$thesumofthedigitswillbe10000.$$$$findhowmanydigitsthisnumber$$$$has.$$

Answered by Rasheed.Sindhi last updated on 17/Oct/19

$$Let1comesxtimes\mathrm{\&}2ytimesin$$$$thenumber.$$$$Thesumofthedigits=2018$$$$x+2y=2018\dots \dots \dots \dots \dots \dots \dots \dots A$$$$Thenumberwillequal:$$$$1({10}^{u_1}+{10}^{u_2}+\dots +{10}^{u_x})$$$$+2({10}^{v_1}+{10}^{v_2}+\dots +{10}^{v_y})$$$$5timesthenumber:$$$$5({10}^{u_1}+{10}^{u_2}+\dots +{10}^{u_x})$$$$+10({10}^{v_1}+{10}^{v_2}+\dots +{10}^{v_y})$$$$=5({10}^{u_1}+{10}^{u_2}+\dots +{10}^{u_x})$$$$+1({10}^{v_1+1}+{10}^{v_2+1}+\dots +{10}^{v_y+1})$$$$Thesumofthedigitsnow:$$$$5x+y=10000\dots \dots \dots \dots \dots \dots .B$$$$FromA\mathrm{\&}B:$$$$x=1998\mathrm{\&}y=10$$$$\therefore Thenumberofdigitsinthenumber$$$$=x+y=1998+10=2008$$

Commented by mr W last updated on 17/Oct/19

$$thanks!$$

Answered by MJS last updated on 17/Oct/19

$$m+2n=2018\left(1\right)$$$$1\times 5=52\times 5=10\Rightarrow$$$$5m+n=10000\left(2\right)$$$$\Rightarrow m=1998\wedge n=10$$$$\Rightarrow \mathrm{it}\mathrm{has}2008\mathrm{digits}$$

Commented by mr W last updated on 17/Oct/19

$$thanks!$$

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