Write 1676 as a product of prime numbers only.

Step-by-step solution

1. 1

   Start with the smallest prime. $1676$ is even, so divide by $2$:

   $1676=2\times 838$

2. 2

   Keep dividing while the quotient stays even. $838$ is still even:

   $838=2\times 419\quad\Rightarrow\quad 1676=2^2\times 419$

   $419$ is odd, so no further factor of $2$ exists.

3. 3

   Decide how far to test 419. A composite number always has a prime factor no larger than its square root, and $\sqrt{419}\approx 20.47$. So it is enough to test the primes $2,3,5,7,11,13,17,19$ — eight checks, not four hundred.

4. 4

   Run the trial divisions. $419$ is odd (not $2$); its digit sum is $4+1+9=14$, not a multiple of $3$; it does not end in $0$ or $5$; and $419=7(59)+6=11(38)+1=13(32)+3=17(24)+11=19(22)+1$. None divides evenly, so 419 is prime.

5. 5

   State the factorization.

   $1676=2^2\times 419$

   Check: $4\times 419=1676$ ✓. Since both bases are prime, this is the complete prime factorization, and by the fundamental theorem of arithmetic it is the only one.