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

let us assume two digits be PQ.

As per the condition given in the question,

Product of 2 digits is 18, PQ = 18 … [equation (i)]

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

PQ – 63 = QP … [equation (ii)]

Now, assume two-digit number be PQ, means P = 10p (as it comes in tens digit)

Then,

PQ – 63 = QP

10p + q – 63 = 10q + p

By transposing we get,

9p – 9q – 63 = 0

Divide both side by 9,

P – Q – 7 = 0 … [equation (iii)]

So, PQ =18

P = 18/Q

Substitute the value of P in equation (iii),

(18/Q) – Q – 7 = 0

18 – Q2 – 7Q = 0

Q2 + 7Q – 18 = 0

Q2 + 9Q – 2Q – 18 = 0

Q(Q + 9) – 2(Q + 9) = 0

Q + 9 = 0 and Q – 2 = 0

Q = -9 and Q = 2

Hence, Q = 2 [Since value of Q cannot be negative]

Then, P = 18/Q

P = 18/2

P = 9

Hence, the number is 92

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