Prove that every integer ending in 3 or 7 has a prime factor that also ends in 3 or 7

Question

Prove that every integer ending in 3 or 7 has a prime factor that also ends in 3 or 7.

I have that such an integer n has n=3 or 7(mod 10) but don't know where to go from there.

Then show that there are infinitely many prime numbers n with n=3 or 7 (mod 10)

Try to prove the contrapositive statement: if a positive integer $n$ does not have a prime factor that ends in $3$ or $7$ (i.e. prime factors are $2, 5$ or $p \equiv \pm 1 \pmod{10}$) then $n$ does not end in $3$ or $7$. — Ivan Loh, Oct 16 '13 at 15:14
There are only a few digits that factors of $n$ can end in. What are they? — Empy2, Oct 16 '13 at 15:15
Well, of the four primes that are digits, only $;3;;and;;7;$ have multiples that end with $;3,,,7;$ . Add to that the prime $;11;$ and we're done... — DonAntonio, Oct 16 '13 at 15:30

lab bhattacharjee · Accepted Answer · 2013-10-16T15:47:40.717

4

HINT:

First of all, any integer$(N)$ ending in $3$ or $7$ will not be divisible by $2,5$(why?)

So, the factors of $N$ must be $\equiv1,3,7,9\pmod {10}$

Observe that $1\cdot1 \equiv1, 9\cdot9\equiv1, 1\cdot9\equiv9 \pmod{10}$

So, the product of the primes(or their powers) $\equiv1,9\pmod{10}$ will be $\equiv1$ or $9\pmod{10}$

edited Oct 16 '13 at 15:47

answered Oct 16 '13 at 15:42

lab bhattacharjee

274,582

Surely you meant $1 \cdot 9 \equiv 9 \pmod{10}$. Fix it, please. – Smylic Oct 16 '13 at 15:45
@Smylic, thansk for your observation.Fixed – lab bhattacharjee Oct 16 '13 at 15:48
I have just added a second part, do you have an answer for it? – Rachel Oct 19 '13 at 11:07
@Rachel, http://math.stackexchange.com/questions/115121/primes-ending-in-7 and http://goo.gl/GocOSO. But, in general you can create a new problem to draw attention of more audience – lab bhattacharjee Oct 19 '13 at 12:55

score 1 · Answer 2 · answered Sep 22 '16 at 16:05

Every prime number is congruent to $1,3,7$, or $9$ modulo $10$. Here is a a multiplication table modulo $10$

\begin{array}{c|cccc} \times & 1 & 3 & 7 & 9 \\ \hline 1 & 1 & 3 & 7 & 9 \\ 3 & 3 & 9 & 1 & 7 \\ 7 & 7 & 1 & 9 & 3 \\ 9 & 9 & 7 & 3 & 1 \\ \end{array}

Note that every integer ending in $3$ or $7$ had at least one divisor that ends in $3$ or $7$. Since a number can only have a finite number of divisors, it follows that at least one prime divisor must end in $3$ or $7$.

Since $\gcd(10,3) = \gcd(10,7) = 1$, Dirichet's theorem says that every arithmetic progression of the form $10n+3$ and $10n+7$ will contain infinitely many prime numbers.

score 1 · Answer 3 · answered Sep 22 '16 at 16:13

1

Maybe "prime" is a Red Herring here. Show more generally: if a number ending in $3$ or $7$ is factored in any way (prime or not) then at least one of the factors ends in $3$ or $7$.

answered Sep 22 '16 at 16:13

GEdgar

111,679

score 0 · Answer 4 · answered Sep 22 '16 at 16:34

0

No product of two elements in $\{1,5,9\}\subset{\mathbb Z}_{10}$ is in $\{3,7\}$.

answered Sep 22 '16 at 16:34

Christian Blatter

226,825

Prove that every integer ending in 3 or 7 has a prime factor that also ends in 3 or 7

4 Answers4