Algebra???

Algebra Level 4

A two-digit number is multiplied by twice the sum of its digits. If the new number formed is 1558 greater than the original number. What was the original number?

The answer is 82.

2 solutions

Arpan Konar
Oct 4, 2015

Let the number be N=10x+y. Hence,N $\times 2(x+y)=N+1558$ Or,N= $\frac {1558}{2x+2y-1}$ The factors of 1558 are 1,2,19,38,41,82,779,1558. Since $0 \leq x \leq 9$ & $0 \leq y \leq 9$ therefore $-1 \leq 2x+2y-1 \leq 35$ . Therefore 2x+2y-1 can only be 1,2 or 19 for the left hand side of the equation to be an integer.But when it takes values 1 &2 then N=1558 &779 respectively(which are rejected as N is a two digit number).Hence only 2x+2y-1=19 is possible,then N= $\frac {1558}{19}$ =82(Ans)

best of luck for kvpy!

shuvam keshari - 5 years, 7 months ago

Thanx....you didnt appear this year?

Arpan Konar - 5 years, 7 months ago

no, but I solved the question paper on 2nd November at home. mathematics was really good.

shuvam keshari - 5 years, 7 months ago

how was your exam??

shuvam keshari - 5 years, 7 months ago

@Shuvam Keshari – Good....could have been better...i should have attempted physics in section 2 instead of maths

Arpan Konar - 5 years, 7 months ago

how was your Olympiad exams??

shuvam keshari - 5 years, 6 months ago

Astronomy was good,in chemistry i got many wrong as i made many guess,and physics was ok...how was yours?

Arpan Konar - 5 years, 6 months ago

same here. not much chance in chem!

shuvam keshari - 5 years, 6 months ago

@Shuvam Keshari – I think i only have got chance in astro...what do u think will be the cutoff for each?

Arpan Konar - 5 years, 6 months ago

@Arpan Konar – i think AT MAX, phy-115, chem-150, astro-150

shuvam keshari - 5 years, 6 months ago

@Arpan Konar – so, how was your Olympiad exams??

shuvam keshari - 5 years, 5 months ago

@Shuvam Keshari – 95 in nsep,119 in nsec and 180 in nsea....and urs?

Arpan Konar - 5 years, 5 months ago

@Arpan Konar – 100 in nsep, 161 in nsea and 91 nsec.

physics was much less than expected, and i think i need to start studying chem real hard:(

shuvam keshari - 5 years, 5 months ago

@Arpan Konar – you know what, i think, it is useless talking on the 'comments' section of this problem. in future join me in google hangouts!!

shuvam keshari - 5 years, 5 months ago

@Arpan Konar – btw, happy new year and see you on 30th january!

shuvam keshari - 5 years, 5 months ago

Shuvam Keshari
Oct 4, 2015

let the number be represented as 10x+y

by framing the conditions in the form of an equation, we get a hyperbola and we may look for the lattice points.

here x can take values-- 1,2,3,4,5,6,7,8,9. we find that only for x=8, we get a reasonable value of y=2

Algebra???

The answer is 82.

2 solutions

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