Write 167/42 as a mixed number in lowest terms.

Step-by-step solution

1. 1

   Check whether the fraction reduces. Factor the denominator: $42=2\times3\times7$. So the fraction reduces only if $167$ is divisible by $2$, $3$ or $7$. It is odd (not $2$), its digit sum is $1+6+7=14$ which is not a multiple of $3$, and $7\times23=161$ with $167-161=6\neq0$. In fact $167$ is prime, so

   $\gcd(167,42)=1$

   and the fraction is already in lowest terms ✓.

2. 2

   Divide to find the whole-number part. Because $167>42$, the fraction is improper and contains at least one whole:

   $42\times3=126,\qquad42\times4=168>167$

   so the quotient is $3$ — four would already overshoot by $1$.

3. 3

   Find the remainder.

   $167-126=41$

   Since $41<42$, this is a valid remainder and the fractional part is $\tfrac{41}{42}$.

4. 4

   Assemble the mixed number.

   $\frac{167}{42}=3\frac{41}{42}$

   Check by reversing: $3\times42+41=126+41=167$ ✓.

5. 5

   Note how close it is to 4. The fractional part $\tfrac{41}{42}\approx0.976$ is just one forty-second short of a whole, so $\tfrac{167}{42}\approx3.976$ — barely below $4$, consistent with $42\times4=168$ being only $1$ more than $167$. As a decimal it repeats, since $42$ contains the primes $3$ and $7$.