the digit of three digit national number are in ap and there's sum is 18. The number obatained by reversing the digits is 594 less than the original number. find the original number.
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Given a three digit number 100x + 10y + z, where it's digits are in AP.
=> x, y, z
And,
x + y + z = 18
Here, our number of terms is 3
Sum of terms in an AP = (n/2) [a + l]
where a is the first term, and l is the last term of the AP.
Hence,
18 = 1.5 [x+z]
=> x+z = 12... (i)
=> The number is 12 + y
Hence,
12+y = 18
=> y = 6
Now, when the digits are reversed, the number is:
100z + 10y + x
=> 100z + 60 + x = 100x + 60 + z - 594
=> 99z - 99x = -594... divide by 99
=> z - x = -6
=> z = 6 + x
Substitute z in (i)
=> x+x = 6
=> x = 3
So,
3 + 6 + z = 18
=> z = 9
=> The number is: 100(3) + 10(6) + 9 = 369
Hope this helped.
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