Arithmetic Progression on Digits

Algebra Level 2

The digits of a three-digit positive number forms an arithmetic progression with a common difference of 1. The sum of the digits is equal to the product of the first and second digits. What is the number?


The answer is 345.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

the digits are x , x + 1 , x + 2 x, x+1,x+2

x + ( x + 1 ) + ( x + 2 ) = x ( x + 1 ) x+ (x+1) + (x+2) = x (x+1)

⟹ 3 x + 3 = x 2 + x \implies 3x + 3 = x^2 + x

⟹ x 2 − 2 x − 3 = 0 \implies x^2 - 2x - 3 = 0

⟹ x = 3 \implies x = 3

⟹ x + 1 = 4 \implies x+1 = 4

⟹ x + 2 = 5 \implies x+2 = 5

the number is 345

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Why can't it be 321? The common difference is still 1.

Saya Suka - 4 years, 7 months ago

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common difference is second term minus first term or third term minus first term. the common difference in 321 is -1.

A Former Brilliant Member - 4 years, 7 months ago

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