isi-entrance 2006 Q9

isi-entrance · India · solved Number Theory Properties of Integer Sequences and Digit Analysis
Find a 4-digit number $N$ such that $N = 4M$, where $M$ is the number obtained by reversing the digits of $N$.
$N = 1000a + 100b + 10c + d$, $M = 1000d + 100c + 10b + a$.
Since $M \leq 2499$, we have $d = 1$ or $2$, $c = 1,2,3,4$.
$1000a + 100b + 10c + d = 4(1000d + 100c + 10b + a)$ $\Rightarrow 1333d + 130c - 20b - 332a = 0$.
Putting $d=2, c=1$: $2796 - 332a = 20b$. For LHS to be a multiple of 20 and less than 200, $a=8$: $2796 - 2656 = 140 = 20b \Rightarrow b=7$.
Therefore $N = 8712$, $M = 2178$. 
Find a 4-digit number $N$ such that $N = 4M$, where $M$ is the number obtained by reversing the digits of $N$.
Paper Questions
Q1 Q2 Q3 Q4 Q5 Q7 Q8 Q9 Q10