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.
68
Understanding Divisibility by 3
The question asks us to find the sum of squares of the greatest and smallest possible values of the digit K such that the number 45082K is divisible by 3. A key concept here is the divisibility rule for 3.
A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
Calculating the Sum of Digits
The given number is 45082K. This number has six digits, where K is the digit in the units place.
Let's find the sum of the known digits:
Sum of known digits = 4 + 5 + 0 + 8 + 2 = 19.
The sum of all digits in the number 45082K is the sum of the known digits plus the digit K. So, the total sum of digits is $19 + K$.
Finding Possible Values for K
For the number 45082K to be divisible by 3, the sum of its digits ($19 + K$) must be divisible by 3.
Since K is a single digit, its possible values are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
We need to check which of these values, when added to 19, results in a number divisible by 3.
| Value of K | Sum of digits (19 + K) | Is (19 + K) divisible by 3? | Possible Value of K? |
|---|---|---|---|
| 0 | $19 + 0 = 19$ | No | No |
| 1 | $19 + 1 = 20$ | No | No |
| 2 | $19 + 2 = 21$ | Yes ($21 \div 3 = 7$) | Yes |
| 3 | $19 + 3 = 22$ | No | No |
| 4 | $19 + 4 = 23$ | No | No |
| 5 | $19 + 5 = 24$ | Yes ($24 \div 3 = 8$) | Yes |
| 6 | $19 + 6 = 25$ | No | No |
| 7 | $19 + 7 = 26$ | No | No |
| 8 | $19 + 8 = 27$ | Yes ($27 \div 3 = 9$) | Yes |
| 9 | $19 + 9 = 28$ | No | No |
The possible values for K that make the number 45082K divisible by 3 are 2, 5, and 8.
Identifying Smallest and Greatest Values of K
From the possible values {2, 5, 8}:
- The smallest value of K is 2.
- The greatest value of K is 8.
Calculating the Sum of Squares
The question asks for the sum of squares of the greatest value and the smallest value of K.
Smallest value squared = $2^2 = 2 \times 2 = 4$.
Greatest value squared = $8^2 = 8 \times 8 = 64$.
Sum of squares = Smallest value squared + Greatest value squared
Sum of squares = $4 + 64 = 68$.
Therefore, the sum of squares of the greatest value and the smallest value of K is 68.
Step-by-Step Solution Summary
- Understand the divisibility rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Calculate the sum of the known digits in the number 45082K: $4 + 5 + 0 + 8 + 2 = 19$.
- Determine the expression for the sum of all digits: $19 + K$.
- Identify the possible single-digit values for K (0 through 9).
- Test each value of K to see if $19 + K$ is divisible by 3. The possible values are 2, 5, and 8.
- Identify the smallest possible value of K (2) and the greatest possible value of K (8).
- Calculate the square of the smallest value: $2^2 = 4$.
- Calculate the square of the greatest value: $8^2 = 64$.
- Calculate the sum of these squares: $4 + 64 = 68$.
Revision Table: Divisibility Rules and K
| Concept | Explanation | Application to 45082K |
|---|---|---|
| Divisibility by 3 | Sum of digits is divisible by 3. | Sum of digits $19 + K$ must be divisible by 3. |
| Possible Values of K | K is a single digit (0-9). | Tested values 0, 1, ..., 9. |
| Values of K for Divisibility | Values of K that make $19 + K$ divisible by 3. | K = 2, 5, 8. |
| Smallest Value of K | Minimum K from possible values. | 2 |
| Greatest Value of K | Maximum K from possible values. | 8 |
| Sum of Squares | (Smallest K)$^2$ + (Greatest K)$^2$. | $2^2 + 8^2 = 4 + 64 = 68$. |
Additional Information: More on Divisibility Rules
Understanding divisibility rules is very helpful in number theory and arithmetic problems. Here are a few other common divisibility rules:
- Divisibility by 2: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8).
- Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
- Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- Divisibility by 10: A number is divisible by 10 if its last digit is 0.
These rules can simplify checking if large numbers are divisible by smaller numbers without performing long division.
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 ?
How many composite numbers are there from 53 to 97 ?
Let x be the least number which when subtracted from 10424 gives a perfect square number. What is the least number by which x should be multiplied to get a perfect square?
