Understanding Number Divisibility

The question asks us to determine which of the given numbers (5, 11, 16, or 8) completely divides the number 1254216. To solve this, we will examine the divisibility rules for each of the options and apply them to the number 1254216.

Checking Divisibility by 5

A number is divisible by 5 if its last digit is either 0 or 5.

The number we are checking is 1254216. The last digit is 6.

Since the last digit is 6, and not 0 or 5, the number 1254216 is not divisible by 5.

Checking Divisibility by 11

A number is divisible by 11 if the difference between the sum of its digits at odd places (from the right) and the sum of its digits at even places (from the right) is either 0 or a multiple of 11.

For the number 1254216:

• Digits at odd places (1st, 3rd, 5th, 7th from right): 6, 2, 5, 1
• Sum of digits at odd places: \(6 + 2 + 5 + 1 = 14\)
• Digits at even places (2nd, 4th, 6th from right): 1, 4, 2
• Sum of digits at even places: \(1 + 4 + 2 = 7\)

The difference between the sums is \(14 - 7 = 7\).

Since the difference, 7, is neither 0 nor a multiple of 11, the number 1254216 is not divisible by 11.

Checking Divisibility by 16

A number is divisible by 16 if the number formed by its last four digits is divisible by 16. This rule is derived from \(16 = 2^4\).

For the number 1254216, the last four digits form the number 4216.

Now, we check if 4216 is divisible by 16:

Divide 4216 by 16:

\(4216 \div 16\)

We can perform long division or repeated subtraction.

\(4216 = 16 \times 200 + 1016\)

\(1016 = 16 \times 60 + 56\)

\(56 = 16 \times 3 + 8\)

So, \(4216 = 16 \times (200 + 60 + 3) + 8 = 16 \times 263 + 8\).

Since there is a remainder of 8 when 4216 is divided by 16, 4216 is not divisible by 16. Therefore, the number 1254216 is not divisible by 16.

Checking Divisibility by 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8. This rule is derived from \(8 = 2^3\).

For the number 1254216, the last three digits form the number 216.

Now, we check if 216 is divisible by 8:

Divide 216 by 8:

\(216 \div 8\)

\(216 = 8 \times 27\)

Since 216 is exactly divisible by 8 with no remainder (\(216 \div 8 = 27\)), the number 1254216 is divisible by 8.

Conclusion

Based on our checks using the divisibility rules:

• 1254216 is not divisible by 5.
• 1254216 is not divisible by 11.
• 1254216 is not divisible by 16.
• 1254216 is divisible by 8.

Therefore, among the given options, the number 1254216 is divisible by 8.

Revision Table: Divisibility Rules

Additional Information: More Divisibility Checks

Understanding divisibility rules helps in quickly determining factors of numbers without performing long division. Here are a few more common divisibility rules:

• Divisibility by 2: The number is even (last digit is 0, 2, 4, 6, or 8). 1254216 ends in 6, so it's divisible by 2.
• Divisibility by 3: The sum of the digits is divisible by 3. Sum of digits of 1254216 is \(1+2+5+4+2+1+6 = 21\). Since 21 is divisible by 3, 1254216 is divisible by 3.
• Divisibility by 4: The number formed by the last two digits is divisible by 4. The last two digits are 16. Since 16 is divisible by 4 (\(16 \div 4 = 4\)), 1254216 is divisible by 4.
• Divisibility by 6: The number is divisible by both 2 and 3. Since 1254216 is divisible by both 2 and 3, it is divisible by 6.
• Divisibility by 9: The sum of the digits is divisible by 9. Sum of digits is 21. 21 is not divisible by 9, so 1254216 is not divisible by 9.
• Divisibility by 10: The last digit is 0. 1254216 ends in 6, so it's not divisible by 10.