The overused number

Number Theory Level 2

I have a 2-digit positive integer, I then reverse the digits to form another 2-digit positive integer.

The product of these 2 positive integers is 1729.

What is the sum of these 2 positive integers?


The answer is 110.

Pi Han Goh by Pi Han Goh , 30, Malaysia

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3 solutions

Venkatachalam J
Apr 18, 2017

Factor of 1729 in pairs as follows:

1 x 1729, 7 x 247, 13 x 133, 19 x 91

The only possible combination for the given problem is 19 x 91

Hence the result =19+91=110

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Anthony Holm
Jan 18, 2017

The prime factors of 1729 are 7, 13, and 19. The only way to multiply two of them to get a 2 digit number is 7*13=91. This is 19 with the digits reversed, so these two are our two numbers and their sum is simply 110.

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Md Zuhair
Jan 17, 2017

Lets say the unit digit of the number be x x and tens digit be y y .

The number is 10 y + x 10y+x , and reversed number y + 10 x y+10x .

Now Product of these two

( y + 10 x ) ( 10 y + x ) = 1729 (y+10x)(10y+x) = 1729 Hence ( y + 10 x ) ( 10 y + x ) = 91 × 19 (y+10x)(10y+x) = 91 \times 19 ( Can be arranged like this)

hence The numbers are 91 91 and 19 19

Adding 91 + 19 = 110 91+19 = 110 (Ans)

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Almost complete. You need to show that those 2 numbers must be 91 and 19 (and cannot be any other 2 numbers).

Pi Han Goh - 4 years, 5 months ago

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