Q) A 2-digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.

Ans:

Step 1: Let’s consider X and Y are the digits of the given number.

Hence the given number is 10 X + Y

∴ the product of the digits = X Y

By 1st given condition: X Y = 18 ………… (i)

Step 2: Next, by the 2nd given condition, when 63 is subtracted the number, its digits are interchanged.

∴ (10 X + Y) – 63 = (10 Y + X)

∴ 9 X – 9 Y = 63

∴ X – Y = 7

∴ X = Y + 7 …………… (ii)

Step 3: Next, solve equations (i) and (ii) to get values of X and Y.

By substituting the value of X from equation (ii) in equation (i) and (ii), we get:

X Y = 18

∴ (Y + 7) Y = 18

∴ Y ² + 7 Y = 18

∴ Y ² + 7 Y – 18 = 0

∴ Y ² + 9 Y – 2 Y – 18 = 0

∴ Y (Y + 9) – 2 ( Y + 9) = 0

∴ ( Y – 2) ( Y + 9) = 0

∴ Y = 2 and Y = – 9

Here we reject Y = – 9 as a digit in a number cannot be negative, and accept Y  = 2

Step 4: from equation (ii), we have:

X = Y + 7

∴ X = 2 + 7

∴ X = 9

Hence, the original number is:  10 X + Y = 10 (9) + 2 = 92

Therefore, the given number is 92.

Check:

1) Product of the digits is 9 x 2 = 18…. condition matched !

2) When we subtract 63 from it, we get 92 – 63 = 29, and we can see that the digits are interchanged… condition matched!

Hence, our solution is correct.

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