Switching the ones digits

Algebra Level 3

I have a pair of two-digit numbers. If I switch their ones digits, the product doesn't change.

Ex:

2 5 × 2 6 = 650 2\textcolor{#D61F06}{5}\times2\textcolor{#D61F06}{6}=650

2 6 × 2 5 = 650 2\textcolor{#D61F06}{6}\times2\textcolor{#D61F06}{5}=650

How many pairs of two-digit numbers are there that have this property and don't share their tens or ones digits? For example, the pair above doesn't work.


The answer is 0.

An Phạm by An Phạm , 16, Vietnam

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1 solution

Steven Yuan
Jan 28, 2018

Let the two numbers be 10 A + a 10A + a and 10 B + b . 10B + b. By the problem conditions, we must have

( 10 A + a ) ( 10 B + b ) = ( 10 A + b ) ( 10 B + a ) 100 A B + 10 A b + 10 B a + a b = 100 A B + 10 A a + 10 B b + a b 10 A b − 10 A a = 10 B b − 10 B a 10 A ( b − a ) = 10 B ( b − a ) . \begin{aligned} (10A + a)(10B + b) &= (10A + b)(10B + a) \\ 100AB + 10Ab + 10Ba + ab &= 100AB + 10Aa + 10Bb + ab \\ 10Ab - 10Aa &= 10Bb - 10Ba \\ 10A(b - a) &= 10B(b - a). \end{aligned}

We cannot have b − a = 0 , b - a = 0, since that means the units digit of both numbers will be the same. However, if b − a ≠ 0 , b - a \neq 0, then we must have A = B , A = B, which is also not possible since the tens digits must be different. There are 0 \boxed{0} pairs of two-digit numbers that satisfy the problem criteria.

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