Question

1200222 ÷ 405

Explanation:

Step1: Simplify the division

We can simplify the division \(1200222\div405\) by dividing numerator and denominator by their greatest common divisor (GCD). First, find the GCD of 1200222 and 405.
The prime factorization of 405 is \(405 = 5\times81=5\times3^4\).
Check if 3 divides 1200222: \(1 + 2+0 + 0+2 + 2+2=9\), which is divisible by 3, so 3 is a factor.
Divide 1200222 by 3: \(1200222\div3 = 400074\)
Divide 405 by 3: \(405\div3 = 135\)
Now we have \(400074\div135\)
Again, 135 and 400074 are divisible by 3: \(400074\div3 = 133358\), \(135\div3 = 45\)
Now we have \(133358\div45\)

Step2: Perform the division

\(133358\div45=\frac{133358}{45}\approx2963.5111\) (or as a fraction \(133358\div45 = 2963\frac{23}{45}\))
Alternatively, we can do the long division directly:
\(405\times2963 = 405\times(3000 - 37)=405\times3000-405\times37 = 1215000-14985 = 1199015\)
\(1200222 - 1199015=1207\)
\(1207\div405\approx2.98\)
So total is \(2963 + 2.98\approx2965.98\)? Wait, no, my first simplification had an error. Let's correct the GCD calculation.
Wait, initial step: 1200222 and 405. Let's use the Euclidean algorithm.
\(1200222\div405 = 2963\) with a remainder: \(405\times2963 = 405\times(2900 + 63)=405\times2900+405\times63 = 1174500+25515 = 1199015\)
\(1200222 - 1199015 = 1207\)
\(1207\div405 = 2\) with remainder \(1207 - 810 = 397\)
\(405\div397 = 1\) with remainder \(8\)
\(397\div8 = 49\) with remainder \(5\)
\(8\div5 = 1\) with remainder \(3\)
\(5\div3 = 1\) with remainder \(2\)
\(3\div2 = 1\) with remainder \(1\)
\(2\div1 = 2\) with remainder \(0\). So GCD is 1. So my initial simplification was wrong. Let's do long division properly.
\(405\) into \(1200222\):
\(405\times2963 = 405\times2000=810000\), \(405\times900 = 364500\), \(405\times60 = 24300\), \(405\times3 = 1215\)
Sum: \(810000+364500 = 1174500+24300 = 1198800+1215 = 1199015\)
Subtract from 1200222: \(1200222 - 1199015 = 1207\)
Now, \(1207\div405 = 2\) (since \(405\times2 = 810\)), remainder \(1207 - 810 = 397\)
So the result is \(2963 + 2+\frac{397}{405}=2965+\frac{397}{405}\approx2965.9802\)

Answer:

\(\approx2965.98\) (or as a fraction \(2965\frac{397}{405}\))