10^a*K + P is divisible by 81. If K, P, a and b are positive integers

If \(10^a*k + p\) is divisible by \( 81\) then it is also divisble by \(3\)

If \(10^a*k + p\) is divisble by 3 then \(10^b*k + p\) is also divisble by 3 ...(I)

Note : the value of b doesn't matter as it only adds zero's and does not alter the divisibilty by \(3\)

e.g. \(81, 801, 8001\) etc are all divisble by \(3\)

If \(10^b*k + p\) is divisible by \(15\) then it is divisible by \(3\) and \(5 \)

We already know \(10^b*k+ p\) is divisible by 3 from ..(I)

All we need to know is whether \(10^b*k + p\) is divisble by \(5\).

\(10^b*k + p\) will be divisble by \(5\) only when p is \(5\) or \(5x\)

eg. \(10+5 =15, 20+10=30, 40+5 =45\) etc

(1) P is the lowest prime which can be expressed in the form of 6n-1, where n is a positive integer.

This tells us p = \(5 \)

Hence \(10^b*k + p\) is a multiple of \(3\) that also ends with a \(5\)

\(10+5 = 15 \)

\(100+5=1005\)

\(40+5 = 45\)

\(400+5 =405\)

etc.

All such numbers are divisible by \(15\)

SUFF.

(2) K is one less than P

This doesn't tell us the value of \( p \)

\(10^2*4+5=405\) Yes
\(10^2*1+2= 102\) No

INSUFF

Ans A

Hope it helped.

But 102 isn't satisfying the constraint in the question stem .
Bunuel Hi , please show some light on this . I can't find any number other than 405 that would fit into this constraint. If that's the case then answer would be D.

Dear conqueror98,

This was not an easy question and I perfectly understand your reason for getting confused.

Notice the two terms in the stem :

\(10^a*K + P\) ...(i)
\(10^b*K + P\) ...(ii)

Are they same? NO.

In (i) power of \(10\) is \(a\) and in (ii) power of \(10\) is \(b \)

Additionally:

First part talks about \(10^a*K + P\) being divisible by \(81 \)
Well as second part is asking whether \(10^b*K + P\) is divisible by \(15.\)

Does the stem say \(10^b*K + P\) is divisible by 81? NO.

So when you say \(102\) does not satisfy the constraint in the stem , it is not correct.

\(102\) does not need to be divisible \(81,\) but it definitely needs to be divisible by \(3.\)

We are trying to find whether \(10^b*K + P\) is divisible by \(15.\)

Now if you go through my first explanation, slowly. I am sure things should be clear.

If anything still remains unclear. Please let me know. Thanks.