Palindromes Formed by Multiplying by 55

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Theorem

$55$ multiplied by any of the odd integers between $91$ and $109$ inclusive produces a palindromic number.


Proof

We have that:

$55 \times 91 = 5005$

Then we have that:

$55 \times 2 = 110$


Thus:

$\forall k: 1 \le k \le 9: 55 \times 2 k = 110k = 110, 220, 330, \ldots, 990$


Thus for $1 \le k \le 9$:

$55 \times \paren {91 + 2 k} = \sqbrk {5kk5}_{10}$


That is:

\(\ds 55 \times 91\) \(=\) \(\ds 5005\)
\(\ds 55 \times 93\) \(=\) \(\ds 5115\)
\(\ds 55 \times 95\) \(=\) \(\ds 5225\)
\(\ds 55 \times 97\) \(=\) \(\ds 5335\)
\(\ds 55 \times 99\) \(=\) \(\ds 5445\)
\(\ds 55 \times 101\) \(=\) \(\ds 5555\)
\(\ds 55 \times 103\) \(=\) \(\ds 5665\)
\(\ds 55 \times 105\) \(=\) \(\ds 5775\)
\(\ds 55 \times 107\) \(=\) \(\ds 5885\)
\(\ds 55 \times 109\) \(=\) \(\ds 5995\)

$\blacksquare$


Sources