In a two-digit number, the unit digit is 6 more than the tens digit, and the product of the two digits is 12 less than the number itself. What could the number be?

Q.In a two-digit number, the unit digit is 6 more than the tens digit, and the product of the two digits is 12 less than the number itself. What could the number be?

Let the tens digit of the two-digit number be \(x\). ∴ The units digit is \((x + 6)\). So, the number is \(10x + (x + 6)\). And the product of the digits is \(x(x + 6)\). ∴ According to the question: \[ x(x + 6) = 10x + (x + 6) - 12 \] \[ x^2 + 6x = 10x + x + 6 - 12 \] \[ x^2 + 6x = 11x - 6 \] \[ x^2 + 6x - 11x + 6 = 0 \] \[ x^2 - 5x + 6 = 0 \] \[ x^2 - 3x - 2x + 6 = 0 \] \[ x(x - 3) - 2(x - 3) = 0 \] \[ (x - 3)(x - 2) = 0 \] ∴ Either \((x - 3) = 0\) or \((x - 2) = 0\) When \((x - 3) = 0\), then \(x = 3\); the number is \(10 \times 3 + (3 + 6) = 39\) When \((x - 2) = 0\), then \(x = 2\); the number is \(10 \times 2 + (2 + 6) = 28\) ∴ The number could be 39 or 28.

Marks	Less than 10	Less than 20	Less than 30	Less than 40	Less than 50	Less than 60	Less than 70	Less than 80
Number of students	4	16	40	76	96	112	120	125

Marks Obtained	Number of Candidates	Cumulative Frequency (Less than Type)
400–450	20	20
450–500	30	50
500–550	28	78
550–600	26	104
600–650	24	128
650–700	22	150
700–750	18	168
750–800	32	200 = n

Weight (kg)	Less than 38	Less than 40	Less than 42
Number of Students	0	4	6

Less than 44	Less than 46	Less than 48	Less than 50	Less than 52
9	12	28	32	35