If the number 715_423 is divisible by 3 (_ denotes the missing digit in the thousandths place), then the smallest whole number in the place of _ is _______.
2
Divisibility Rule of 3
The divisibility rule for the number 3 states that a whole number is divisible by 3 if the sum of its digits is divisible by 3. This rule is very useful for quickly checking if a number, no matter how large, can be evenly divided by 3 without performing long division.
Identifying the Number and Missing Digit
We are given the number 715_423, where the underscore _ represents a missing digit. According to the question, this missing digit is in the thousands place. We need to find the smallest whole number that can replace this underscore so that the entire number 715_423 becomes divisible by 3.
Summing the Digits
First, let's represent the missing digit with a variable, say \(x\). The number can be written as \(715x423\).
Now, let's find the sum of all the digits in the number 715x423:
- Digits are: 7, 1, 5, \(x\), 4, 2, 3.
- Sum of the known digits: \(7 + 1 + 5 + 4 + 2 + 3\)
- Calculating the sum:
So, the sum of the digits of the number 715_423 is \(22 + x\).
Finding the Smallest Whole Number
For the number 715_423 to be divisible by 3, the sum of its digits, which is \(22 + x\), must also be divisible by 3.
Since \(x\) is a single missing digit, it must be a whole number from 0 to 9. We need to find the smallest whole number for \(x\) that makes \(22 + x\) divisible by 3. Let's test the possible values for \(x\) starting from 0:
| Value of \(x\) (Missing Digit) | Sum of Digits (\(22 + x\)) | Divisible by 3? |
|---|---|---|
| 0 | \(22 + 0 = 22\) | No (22 ÷ 3 is not a whole number) |
| 1 | \(22 + 1 = 23\) | No (23 ÷ 3 is not a whole number) |
| 2 | \(22 + 2 = 24\) | Yes (24 ÷ 3 = 8) |
From the table, we can see that when \(x = 2\), the sum of the digits is \(24\), which is perfectly divisible by 3. Since we started checking from the smallest possible whole number (0) for \(x\), the value 2 is the smallest whole number that satisfies the condition.
Therefore, the smallest whole number that can replace the underscore _ to make the number 715_423 divisible by 3 is 2. The complete number would then be 7,152,423.
Important Questions from Numerical Computation
Ratio of the angles of a triangle are 1 : 2 : 6. What is the smallest angle?
It would take one machine 4 hours to complete a production order and another machine 2 hour to complete the same order. If both machines work simultaneously at their respective constant rates, the time taken to complete the same order is ________ hours.
Two design consultants, P and Q, started working from 8 AM for a client. The client budgeted a total of USD 3000 for the consultants. P stopped working when the hour hand moved by 210 degrees on the clock. Q stopped working when the hour hand moved by 240 degrees. P took two tea breaks of 15 minutes each during her shift, but took no lunch break. Q took only one lunch break for 20 minutes, but no tea breaks. The market rate for consultants is USD 200 per hour and breaks are not paid. After paying the consultants, the client shall have USD_remaining in the budget.
What is the value of \(1 + \frac{1}{4} + \frac{1}{{16}} + \frac{1}{{64}} + \frac{1}{{256}} + \ldots ?\)
A 1.5 m tall person is standing at a distance of 3 m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters?
