Powers of 176 end in 76

14 Feb 2022

Vikram Hegde tweeted a statement about powers of \(176\): they all end in \(76\).

I was about to ask for a proof before I realized I’m procrastinating and could write an R function to test this pretty quickly. So I did, and claimed that the statement was not true. Only the first \(8\) digits end in \(76\). Proof by computer!

But no. These numbers get big pretty fast, and floating point is not great with trailing digits of big numbers. Vikram and Manish noticed some oddities with the sequence. Manish went a step further, noting that powers all all digits ending in \(6\) ought to end in \(6\).

But I’m still procrastinating, so here’s a proof by induction: all powers of \(176\) end in \(76\).

The proof

First, note that \(176^1 = 176\) and \(176^2 = 30976\), both of which end in \(76\).

Now suppose \(176^m\) ends in \(76\). If we can establish that \(176^{m+1}\) ends in \(76\), we’re done.

Since \(176^m\) ends in \(76\), we can write

\[\begin{equation} 176^m = 100c + 76, \end{equation}\]

for some natural number \(c\). So now we have

\[\begin{align} 176^{m+1} &= 176^m 176 \\ &= (100c + 76)(100 + 76) \\ &= 100^2 c + 100 c \cdot 76 + 100 \cdot 76 + 76^2 \\ & \text{defining } 100 c = c_1, 76c = c_2, 76 = c_3, \\ &= 100 (c_1 + c_2 + c_3) + 5776 \\ &= 100 (c_1 + c_2 + c_3) + 5700 + 76 \\ & \text{defining } 57 = c_4, \\ &= 100 (c_1 + c_2 + c_3 + c_4) + 76 \\ & \text{defining } c' = c_1 + c_2 + c_3 + c_4, \\ &= 100 c' + 76, \end{align}\]

which is of the same form as what we had for \(176^m\). So \(176^{m+1}\) must also end in \(76\)…? I’m not entirely sure this follows, since our earlier statement was applied for some \(c\), not all \(c\).

But anyway, assuming it’s true (my favorite way to proof), we have what we need. The base case says \(176^2\) ends in \(76\), and the induction step says if \(176^m\) ends in \(76\) then so does \(176^{m+1}\). I’ll stop here—these are dark powers and I dare not toy with them. Thanks Vikram and Manish for this problem!

(I like proofs by induction because they make me picture a little train engine, chugging along establishing the statement till the end of time.)

EDIT: I changed the factorization from \(176^m = 10c + 76\) to \(176^m = 100c + 76\). I think this makes it a little stronger, since now \(c\) doesn’t need to be a multiple of \(10\) to keep the trailing \(76\). I think this also makes the step I was unsure about go through: \(c\) can be any natural number and \(100 c + 76\) will still end in \(76\).