Find the value of the missing digit that makes the statement true.
214,21_ is divisible by 11
Determine whether the statement is true or false. Why?
If a natural number is divisible by 5 and 2, then it must also be divisible by 10.
214,214 is divisible by 11.
second one: true
To find the missing digit that makes the statement "214,21_ is divisible by 11" true, we need to calculate the sum of the odd-placed digits and the even-placed digits separately, and then find the difference between them.
Let's break down the number:
- Odd-placed digits: 2, 4, _
- Even-placed digits: 1, 1
First, we need to calculate the sum of the odd-placed digits: 2 + _ = 2 + _ = 2_
Next, we need to calculate the sum of the even-placed digits: 1 + 1 = 2
To make the number divisible by 11, the difference between the sum of the odd-placed digits and the sum of the even-placed digits should be a multiple of 11.
Therefore, the missing digit should be the number that, when added to the sum of the odd-placed digits (2_), gives a difference that is a multiple of 11.
We can try different digits as the missing digit to see which one satisfies this condition:
- If the missing digit is 0: 20 - 2 = 18, which is not a multiple of 11.
- If the missing digit is 1: 21 - 2 = 19, which is not a multiple of 11.
- If the missing digit is 2: 22 - 2 = 20, which is a multiple of 11.
Therefore, the missing digit that makes the statement true is 2.
Regarding the statement "If a natural number is divisible by 5 and 2, then it must also be divisible by 10":
This statement is true.
A natural number divisible by both 5 and 2 must be a multiple of the least common multiple (LCM) of 5 and 2, which is 10.
Since 10 is a multiple of both 5 and 2, any natural number divisible by 5 and 2 will also be divisible by 10.