Smallest Multiple of 66 At Least 600,000
Find the smallest multiple of $66$ that is greater than or equal to $600{,}000$.
Hints
- You want $\lceil 600{,}000 / 66 \rceil \times 66$. Start by computing $600{,}000 / 66$.
- Simplify: $600{,}000 / 66 = 100{,}000 / 11$. Do the long division to get $9{,}090.\overline{90}$.
- Round up to $9{,}091$ and multiply: $66 \times 9{,}091 = 594{,}000 + 66 \times 91 = 594{,}000 + 6{,}006 = 600{,}006$.
Worked Solution
How to Think About It: You want $\lceil 600{,}000 / 66 \rceil \times 66$. The cleanest approach is direct division: divide $600{,}000$ by $66$, round up to the nearest integer, then multiply back. Under interview pressure without a calculator, you can get there in a few steps of mental arithmetic using known multiples of $66$.
Quick Estimate: $66 \times 10{,}000 = 660{,}000$. That is $60{,}000$ too high. $66 \times 9{,}000 = 594{,}000$. That is $6{,}000$ too low. So the answer is between $66 \times 9{,}000$ and $66 \times 10{,}000$, and we need to add enough multiples of $66$ to bridge the $6{,}000$ gap.
Formal Solution:
Divide directly: $$\frac{600{,}000}{66} = \frac{100{,}000}{11} = 9{,}090.909\ldots$$
Round up: $\lceil 9{,}090.909 \rceil = 9{,}091$.
Multiply back: $$66 \times 9{,}091 = 66 \times 9{,}000 + 66 \times 91 = 594{,}000 + 6{,}006 = 600{,}006$$
Verify: $600{,}006 \geq 600{,}000$ and $600{,}006 - 66 = 599{,}940 < 600{,}000$, so $600{,}006$ is indeed the smallest such multiple.
Answer: $600{,}006$.
Intuition
The general recipe for 'smallest multiple of $k$ that is at least $N$' is always $\lceil N/k \rceil \times k$. Mentally, you are rounding $N/k$ up to the next integer. The ceiling function is the right tool here, and it shows up regularly in quant work: computing the number of hedging lots needed to cover a position, rounding up to the next contract size, or determining the minimum number of time steps in a simulation.
The arithmetic shortcut -- recognizing $66 = 6 \times 11$ and simplifying $600{,}000 / 66 = 100{,}000 / 11$ -- is a useful habit. Reducing fractions before doing division keeps the numbers smaller and reduces error. In mental math under pressure, every simplification helps.