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Today Puzzle #741

Puzzle No. 741鈥� Monday 18 May

Find a 2-digit number so that if we multiply its digits together, and then multiply that by 3, we still get the same number?There are exactly two answers to this problem. Can you find both?

Today’s #PuzzleForToday has been set by Dr Nicos Georgiou, a senior mathematics lecturer from the University of Sussex.

Click here for the answer

The two numbers are 15 and 24.

Let the unknown number be XY, with X being the tens digit and Y the unit digit. Then, the number can be written as 10 X + Y. From the problem, we have that

3XY = 10 X + Y.

We also know that X can be 1,2,3, …, 9 and Y can take values 0, 1, …. 9. Now one can go forward with trial and error of all 2dgit numbers, but it will take a lot of time. Here’s a way to do it faster. By moving 10 X to the left side, and factoring out X, the equation becomes

(3Y – 10)X = Y.

Now, 3Y – 10 needs to be positive since the right-hand side of the equation is positive. So Y can only take the values 4, 5, …, 9. Now, if

Y = 4, the equation becomes (3 x 4 - 10) X = 4, so X = 2 and the number is 24.

Y = 5, the equation becomes (3 x 5 - 10) X = 5, so X = 1 and the number is 15.

Y = 6, there is no solution as X needs to be a whole number.

For Y = 7, 8, 9 there are no solutions, because irrespective of the value of X, the left-hand side of the equation is bigger than 9, which is the maximum value of Y.

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