Determine the remainder when 4412 is divided by 7?

44¹² ÷ 7 = (42 + 2)¹²

By using binomial theorem, we have

(42 + 2)¹² = ¹²C0 × 42¹² × 2⁰ + ¹²C1 × 42¹¹ × 2¹ + …………+ ¹²C12 × 42⁰ × 2¹²

So, in the above expression only the last term will not be completely divisible by ‘7’.

So, required remainder = Remainder obtained when ¹²C12 × 42⁰ × 2¹² is divided by ‘7’ or 2¹² is divided by ‘7’.

Now, 2¹² ÷ 7 = (2³)⁴ ÷ 7 = 8⁴ ÷ 7 = (7 + 1)⁴ ÷ 7

So, again by binomial expansion, we get

(7 + 1)⁴ = ⁴C0 × 7¹² × 1⁰ + ⁴C1 × 7¹¹ × 1¹ + …………+ ⁴C4 × 7⁰ × 1⁴

So, last term will not be divisible by ‘7’.

So, required remainder = ⁴C4 × 7⁰ × 1⁴ = 1