This is fun!
Number Theory
Level
3
While doing some operations i came across a number with the following property.
a^{b} X c^{d} =abcd.
Which is the number abcd satisfying the above property. (Remember abcd means the positioning of the numbers a,b,c,d and not multiplication)
The answer is 2592.
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actually question should be a^b × c^a = abca Let a = 1 then a^b = 1^b = 1 and c^a = c so a^b × c^a = 1 × c = c is 1 digit number so a = 1 is not possible.
Let a = 2 2^b as b having values from 1 to 9 as
2^1, 2^2 2^3 2^4 2^5 2^6 2^7 2^8 2^9
2, 4 , 8, 16 , 32 , 64 , 128 , 256 , 512 series 1
and c having values from 1 to 9 so c^a =
1, 4, 9, 16, 25, 49, 64 or 81 series 2 so abca = 2bc2
the number is starting with 2 and having 2 in the unit place. it is possible by multiplication of 32 (2^5) from 1st series and 81 (9^2) from 2nd series.
2^5 × 9^5 = 2592