It isn't a Typo!

Number Theory Level 2

$\begin{aligned} 23 \times 64 = 32 \times 46 \\ 36 \times 42 = 63 \times 24 \\ 26 \times 31 = 62 \times 13 \end{aligned}$

We call a pair of 2-digit positive integers reversible if their product remains the same when we reverse the digits in both integers.

The above shows that the pairs $(23,64), (36,42), (26,31)$ are reversible.

Which of the following is not reversible?

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(34,86)

(14,82)

(12,42)

They are all reversible

2 solutions

Giorgos K.
Feb 24, 2018

Mathematica

Select[{{34, 86}, {12, 42}, {14, 82}},Times @@ IntegerReverse@@#==Times@@#&]

returns {} which means they are all reversible

Leonblum Iznotded
Jul 23, 2018

Concretely, it is enough to calculate each product ; and compare with the "reversed" product (the product of reversed numbers).

But we think "can we compare without calculating ?"

First product : 34x86 shall be compared to 43x68. Remarking the multiples of 2 : 34x86=17x2x2x43. And 43x68=43x2x2x17. They are equal.
Second product : 12x42=12x2x21 shall be compared to 21x24=21x2x12. Equal (no need to decompose totally in prime factors; comparing to divide what we need is enough).
Third product : 14x82=14x2x41 to be compared to 41x28=41x2x14. Equal.

Of course any of us wonders "how many of 2-digit products of 2 numbers exist ? (in decimal system)" ; (same question in hexadecimal system etc) ; a machine can count them, but is there a way to count humanly without trusting any machine? ; is there an easy manner to group pawns, or to fold a hinge between graduated bars, to solve it without paper and without calculator ? (zero watt, zero tree)

It isn't a Typo!

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