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Show that a perfect square cannot end with digits 2,3 or 7?

Note by Goutam Narayan 4 years, 5 months ago

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We look modulo 10.A number can give remainder 0,1,2...9 mod 10.After checking for all of these values, we see that none of them are \(\equiv\) to 2,3 or 7 (mod 10)

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can u plz show it

Well ,when a number \( n=10k+r\), then \(n^2=100k^2 + 20kr + r^2 \equiv r^2 \pmod{10}\).Now for

\(r=1, n^2\equiv 1^2 \equiv 1\pmod {10}\)

\(r=2, n^2\equiv 2^2 \equiv 4 \pmod {10}\)

\(r=3, n^2\equiv 3^2 \equiv 9 \pmod{10}\)

\(r=4, n^2\equiv 4^2 \equiv 6 \pmod{10}\)

\(r=5, n^2\equiv 5^2 \equiv 5 \pmod{10}\)

\(r=6, n^2\equiv 6^2 \equiv 6 \pmod{10}\)

\(r=7, n^2\equiv 7^2 \equiv 9 \pmod{10}\)

\(r=8, n^2\equiv 8^2 \equiv 4 \pmod{10}\)

\(r=9, n^2\equiv 9^2 \equiv 1 \pmod{10}\)

@Bogdan Simeonov – thank u very much///

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Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestWe look modulo 10.A number can give remainder 0,1,2...9 mod 10.After checking for all of these values, we see that none of them are \(\equiv\) to 2,3 or 7 (mod 10)

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can u plz show it

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Well ,when a number \( n=10k+r\), then \(n^2=100k^2 + 20kr + r^2 \equiv r^2 \pmod{10}\).Now for

\(r=1, n^2\equiv 1^2 \equiv 1\pmod {10}\)

\(r=2, n^2\equiv 2^2 \equiv 4 \pmod {10}\)

\(r=3, n^2\equiv 3^2 \equiv 9 \pmod{10}\)

\(r=4, n^2\equiv 4^2 \equiv 6 \pmod{10}\)

\(r=5, n^2\equiv 5^2 \equiv 5 \pmod{10}\)

\(r=6, n^2\equiv 6^2 \equiv 6 \pmod{10}\)

\(r=7, n^2\equiv 7^2 \equiv 9 \pmod{10}\)

\(r=8, n^2\equiv 8^2 \equiv 4 \pmod{10}\)

\(r=9, n^2\equiv 9^2 \equiv 1 \pmod{10}\)

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