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A two digit number is such that the product of its digit is 18.When 63 is subtracted from the number , the digits interchange their places.Find the number.

Snehil Sinha
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4 Answers4

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Alternative solution:

18 may be factorised into products of single digit numbers as 2*9 or 6*3

We know the number is greater than 63, so first digit must be greater than 6.

Only one possibility.

Explanation:

Algebra is very often a sledgehammer in cases like this. The algebraic answer will always work, and is valid whether numbers are integers, reals, or complex (or...) This alternative is just a shortcut which exploits the fact that we are dealing with integers (a UFD) which have unique factorisations, but not real numbers (a field) in which every number is a factor of every other number (except zero).

Because discrete integers and continuous real numbers are quite different, they can have very different properties. So problems which are hard in one domain can be easy in another. For instance finding the log of a number is easy in real numbers, but can be nearly impossible in discrete maths, which is the basis of modern cryptography.

Richard
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A two digit number

Let $n = 10a+b$.

is such that the product of its [digits] is 18.

$ab = 18$

When 63 is subtracted from the number, the digits interchange their places.

$10a+b-63 = 10b+a$

Find the number.

$n =\;?$

Can you see where to go from here?

senshin
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  • I am stuck over here itself.Iderived this much on my own but dont know what to do next. – Snehil Sinha Jun 20 '14 at 09:21
  • Another hint: you know that $a$ and $b$ both have to be numbers from $1, 2, \dots, 9$ (since they are digits). If you don't know how to solve this analytically, do you know how to solve it by trying possible values of $a$ and $b$? (Also, when asking a question on this site, please remember to put all the work you've done so far in your question to make it easier for other users to help you.) – senshin Jun 20 '14 at 09:22
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    You have 2 equations in the two unknowns $a,b$. Solve one equation for one of the unknowns in terms of the other, and then substitute that into the other equation to get an equation in a single unknown. – Gerry Myerson Jun 20 '14 at 09:35
  • @GerryMyerson thanx a lot.I got it. – Snehil Sinha Jun 20 '14 at 09:38
  • Good. Write it up and post it as an answer. – Gerry Myerson Jun 20 '14 at 09:41
  • @GerryMyerson Sure i will do so – Snehil Sinha Jun 20 '14 at 09:57
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Let the number be n and x and y be tens and unit digit so

n=10x+y

now, x*y=18

Also, n-63=10y+x from condition 2

then,

10x+y-63=10y+x

or 9x-9y=63

x-y=7

possible values of x and y - 8,1 & 9,2 which gives us only 9,2 as the possible answer so the number is 92.

If you want to solve using quadratic equations then substitute y by 18/x and solve.

Check: 92-63=29 (numbers interchanged places)

PS: I just joined this forum today, so I am not an expert of formatting. :)

MonK
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Let the number be=xy x*y=18 63-xy=yx

xy may be=

6*3=18 9*2=18

63-63=0 92-63=29

Therefore the number is 92

Yoyo
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