The number 2918245 is divisible by which of the following numbers?
11
Understanding Divisibility Rules for 2918245
To determine which of the given numbers (3, 11, 12, or 9) divides 2918245, we can use the divisibility rules for each number. Let's examine each option one by one.
Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
The number is 2918245. Sum of digits $= 2 + 9 + 1 + 8 + 2 + 4 + 5 = 31$.
Is 31 divisible by 3? No, $31 \div 3 = 10$ with a remainder of 1.
Therefore, 2918245 is not divisible by 3.
Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
From the previous step, the sum of the digits of 2918245 is 31.
Is 31 divisible by 9? No, $31 \div 9 = 3$ with a remainder of 4.
Therefore, 2918245 is not divisible by 9.
Divisibility by 12
A number is divisible by 12 if it is divisible by both 3 and 4.
We have already checked divisibility by 3, and 2918245 is not divisible by 3.
Let's check divisibility by 4. A number is divisible by 4 if the number formed by its last two digits is divisible by 4. The last two digits of 2918245 are 45.
Is 45 divisible by 4? No, $45 \div 4 = 11$ with a remainder of 1.
Since 2918245 is not divisible by either 3 or 4, it is not divisible by 12.
Divisibility by 11
A number is divisible by 11 if the alternating sum of its digits (starting from the rightmost digit, subtracting the second, adding the third, and so on) is divisible by 11 (the result can be 0 or a multiple of 11).
For the number 2918245:
Alternating sum $= 5 - 4 + 2 - 8 + 1 - 9 + 2$
Let's calculate this step-by-step:
$= (5 - 4) + (2 - 8) + (1 - 9) + 2$ $= 1 + (-6) + (-8) + 2$ $= 1 - 6 - 8 + 2$ $= -5 - 8 + 2$ $= -13 + 2$ $= -11$
Is -11 divisible by 11? Yes, $-11 \div 11 = -1$.
Since the alternating sum of the digits of 2918245 is -11, which is a multiple of 11, the number 2918245 is divisible by 11.
Summary of Divisibility Checks
| Divisor | Rule Applied | Result | Divisible? |
|---|---|---|---|
| 3 | Sum of digits (31) | 31 $\div$ 3 $\neq$ Integer | No |
| 9 | Sum of digits (31) | 31 $\div$ 9 $\neq$ Integer | No |
| 12 | Divisible by 3 and 4? (Last two digits 45) | Not divisible by 3, 45 $\div$ 4 $\neq$ Integer | No |
| 11 | Alternating sum of digits (-11) | -11 $\div$ 11 = -1 | Yes |
Based on the divisibility rules, the number 2918245 is divisible by 11.
Revision Table: Key Divisibility Rules
| Divisor | Divisibility Rule | Example |
|---|---|---|
| 2 | Ends in 0, 2, 4, 6, or 8. | 148 (ends in 8) |
| 3 | Sum of digits is divisible by 3. | 345 (3+4+5=12, 12 is div by 3) |
| 4 | Last two digits form a number divisible by 4. | 1716 (16 is div by 4) |
| 5 | Ends in 0 or 5. | 235 (ends in 5) |
| 6 | Divisible by both 2 and 3. | 42 (div by 2 as it's even, 4+2=6, div by 3) |
| 9 | Sum of digits is divisible by 9. | 585 (5+8+5=18, 18 is div by 9) |
| 10 | Ends in 0. | 560 (ends in 0) |
| 11 | Alternating sum of digits is divisible by 11. | 121 (1-2+1=0, 0 is div by 11) |
Additional Information on Number Divisibility
Divisibility rules are shortcuts that help determine if a number is divisible by another number without performing long division. These rules are based on properties of numbers and place value. Understanding these rules can save time in calculations and help in prime factorization, finding common factors, and simplifying fractions.
While the divisibility rules for small numbers (like 2, 3, 5) are simple, rules for larger numbers like 7, 11, 13, etc., can be a bit more complex but are still very useful. For instance, the rule for 11 involving the alternating sum of digits is a very efficient way to check for divisibility by 11, especially for large numbers. Knowing composite number divisibility rules (like 6, 10, 12) involves checking for divisibility by their prime factors. For example, for 12, you check 3 and 4. For 6, you check 2 and 3.
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select the correct answer using the code given below:
