A two-digit number is such that the product of the digits is 14. When 45 is added to the number, then the digits interchange their places. Find the number.

Let the digits be a and b such that the number is 10a + b.

 ab = 14 and 10a + b + 45 = 10b + a

i.e., 9a – 9b = –45

i.e., a – b = –5

(a + b)2 = (a – b)2 + 4ab = 25 + 4(14) = 81

 a + b = 9

 a = 2, b = 7

 The number is 27.

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