Divisibility by 11: Determining the Smallest Whole Number for Given Digits
Understanding the divisibility rules, particularly for the number 11, can be immensely useful in solving mathematical puzzles and problems. In this article, we explore how to determine the smallest whole number that can replace a placeholder in a given number to ensure its divisibility by 11.
The Divisibility Rule for 11
The divisibility rule for 11 involves calculating the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions. If the difference is either 0 or a multiple of 11, the number is divisible by 11.
Solving for a Missing Digit
Consider the number 97215 x 6. We need to find the smallest whole number in place of x such that the entire number is divisible by 11.
Step-by-Step Solution
Identifying Digits: The digits in the number 97215 x 6 are as follows: Odd positions: 9, 2, 5, 6 Even positions: 7, 1, x, 6 Sum of Digits in Odd Positions: 9 2 5 6 22 Sum of Digits in Even Positions: 7 1 x 6 14 x Calculating the Difference: 22 - (14 x) 8 - x Setting the Difference to 0 or a Multiple of 11: 8 - x 0 8 - x 11 (not a valid single digit) Valid Solutions: 8 - x 0 → x 8 8 - x 11 (not a valid single digit)The smallest whole number that satisfies the condition is 7, since 8 - 7 1 and 8 - 7 11 14, which is not a valid solution because x must be a single digit.
Conclusion
The smallest whole number in place of x that makes 97215 x 6 divisible by 11 is 7. This can be concluded by ensuring that the difference between the sums of the odd and even digits is a multiple of 11.
Similar Problems
Let's explore a similar problem to solidify our understanding:
Problem: If the number 96295 y 6 is completely divisible by 11, find the smallest whole number in place of y.
Solution: Sum of Digits in Odd Positions: 9 2 5 6 22 Sum of Digits in Even Positions: 6 9 y 6 21 y Calculating the Difference: 22 - (21 y) 1 - y Setting the Difference to 0 or a Multiple of 11: 1 - y 0 → y 1
The answer is 1, as 1 - 1 0, which is a multiple of 11.
Thus, the smallest whole number in place of y that makes 96295 y 6 divisible by 11 is 1.
Conclusion
Understanding the divisibility rule for 11 is crucial in solving various mathematical problems and puzzles. This rule applies broadly and can be used to verify the divisibility of a number by 11 quickly and efficiently.