Question

fn is a perfect square and n² = k (mod 10) with 0≤k<9, find the possible values of k​

Answer 1

Given that,

n is a perfect square

${n}^{2} \equiv k \: (mod \: 10), \: \: \: 0 \leq k \le 9$

$\frac{ {n}^{2} - k }{10} \in ℤ$

For 0≤k≤9

$Let, n = 10j + i \: \:, \: \: \: \: \: \: \: j \inℤ {}^{ + } \cup \{0 \}$

$\implies i \in \{0,1,4,5,6,9 \}$

${n}^{2} = 100 {j}^{2} +20ji+ {i}^{2} \equiv k \: (mod \: 10)$

${i}^{2} \equiv k \: (mod \: 10)$

${i}^{2} \in \{0,1,16,25,36,81\}$

Then,

$\boxed{k \in \{0,1,5,6 \}}$