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Consider the equation $(123)_5=(x8)_y$ with $x$ and $y$ as unknown. The number of possible solutions is _____ .

ok got it x<y but why y> 8 ? plz expalin ?
since (x8)..one number here is 8 so base has to be more than 8
for base 2 numbers are 0,1
base 3..0,1 2
and similarly if a number is 8...its base has to be more than 8..
thnku

Converting both sides to decimal,

$25+10+3=x$*$y+8$

So, $xy=30$

Possible pairs are $(1,30),(2,15),(3,10)$ as the minimum base should be greater than $8.$

### 1 comment

And also x<y (strictly). Otherwise pair (30,1), (15,2) can also be included.

Changing (123) base 5 into base 10= 1*25+2*5+3*1=38 Changing x8 base y in decimal= x*y+8 Equating both we get xy+8=38

• xy=30
• possible combinations =(1,30),(2,15),(3,10)

but we have ‘8’ present in x8 so base y>8 as all three are satisfying the conditions so total solutions =3 hence ans is ( C) part

why not 5,6 plz someone explain properly ..
8 can't be part of digits if we take  5,6
here the equation is x*y=30 so the possible values for x and y is (5,6) other values like (6,5) (3,10)(10,3) etc are not valid.
by

How can 5,6 be valid when the number with base x is 8?
vlaue of y never less then 8 show (5,6) is also invalid
Can someone explain how we chhose the x and y pairs from xy= 30