Use place values to find a number
is a three-digit positive number that is 26 times the sum of its digits. When the digits of are reversed, the new number is 198 more than . Given that the digits of are 3 consecutive terms of an arithmetic progression, find . marks
[2014/MJC/mid year/1]
Solution
Place-value modelling
is a three-digit positive number that is 26 times the sum of its digits. When the digits of are reversed, the new number is 198 more than . Given that the digits of are three consecutive terms of an arithmetic progression, find .
Hint
Write for digits .
Turn each clue into one equation.
Solve the three equations.
means , not or in the algebraic sense.
and are between 1 and 9 (the reversal is also a three-digit number). The solution satisfies this.
Answer · Q
.
26 times the digit sum
Move everything to one side: , , .
Reversal adds 198
The middle digit cancels: , so .
Digits in AP
Consecutive terms of an arithmetic progression have equal differences.
Solve
From and , the first equation gives , so and . The source also writes the digits as : reversal gives , so , while gives and .
Check
Both conditions hold, and 2, 3, 4 are in AP.
Explore the idea