Divisibility by 14 Rules Explained
To determine if a number is divisible by 14, we need to check if it meets two conditions based on the divisibility rules for its prime factors, 2 and 7. A number is divisible by 14 if and only if it is divisible by both 2 and 7.
Rule for 2
A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8). This means the number must be an even number.
Rule for 7
There isn't a universally simple trick for 7 like there is for 2 or 5. However, one common method is to take the last digit of the number, double it, and subtract it from the remaining part of the number. If the result is divisible by 7, the original number is also divisible by 7. This process can be repeated if needed. For checking, performing direct division is often the most straightforward method.
Number Options Analysis for Divisibility
Let's examine each option provided to see which one is divisible by 14.
| Number | Divisible by 2? | Divisible by 7? | Divisible by 14? |
|---|---|---|---|
| 67,94,266 | Yes (The last digit is 6, which is even) |
Calculation:
We check if $6794266$ is divisible by 7.
$6794266 \div 7 \approx 970609.42$
Since the result is not a whole number, it is No.
|
No |
| 64,36,639 | No (The last digit is 9, which is odd) | N/A (If not divisible by 2, it cannot be divisible by 14) | No |
| 71,32,960 | Yes (The last digit is 0, which is even) |
Calculation:
We check if $7132960$ is divisible by 7.
$7132960 \div 7 \approx 1018994.28$
Since the result is not a whole number, it is No.
|
No |
| 70,48,048 | Yes (The last digit is 8, which is even) |
Calculation:
We check if $7048048$ is divisible by 7.
$7048048 \div 7 = 1006864$
Since the result is a whole number, it is Yes.
|
Yes |
Final Answer Determination
Based on the analysis:
- Option 1 (67,94,266) is divisible by 2 but not by 7.
- Option 2 (64,36,639) is not divisible by 2, so it cannot be divisible by 14.
- Option 3 (71,32,960) is divisible by 2 but not by 7.
- Option 4 (70,48,048) is divisible by both 2 and 7.
Therefore, the number 70,48,048 is the only number among the options that satisfies both conditions and is divisible by 14.
Important Questions from Number System
Consider the following statements :
1. (25)! + 1 is divisible by 26
2. (6)! + 1 is divisible by 7
Which of the above statements is/are correct ?
If the sum S is divided by 8, what is the remainder ?
If the sum S is divided by 60, what is the remainder ?
Find the sum of squares of the greatest value and the smallest value of K in the number so that the number 45082K is divisible by 3.
How many composite numbers are there from 53 to 97 ?
