Find all the possible values of the missing digits represented by *
a 2 * 6,8 *71 8*919 are divisible by 11.
b 396 *5,48675 * ,349 * are divisible by 9.
a) We can find the possible values of the missing digits by applying the divisibility rule of 11. According to the rule, the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions should be divisible by 11.
Let's analyze each number separately:
For the first number, 2 * 6, the sum of the odd-positioned digits is 2 + 6 = 8, and the sum of the even-positioned digits is 0. Therefore, the difference is 8 - 0 = 8, which is not divisible by 11. Hence, there are no possible values for the missing digit represented by *.
For the second number, 8 * 71 8, the sum of the odd-positioned digits is 8 + (1+8) = 17, and the sum of the even-positioned digits is 0 + 7 = 7. Therefore, the difference is 17 - 7 = 10, which is not divisible by 11. Like in the previous case, there are no possible values for the missing digit represented by *.
For the third number, 8*919, the sum of the odd-positioned digits is 8 + 9 = 17, and the sum of the even-positioned digits is 0 + 1 + 9 = 10. Therefore, the difference is 17 - 10 = 7, which is divisible by 11. Thus, the possible values for the missing digit represented by * are 2 and 4, as they would result in the number 821919 and 841919, respectively.
Therefore, the possible values for the missing digits in the given numbers are 2 and 4.
b) We can find the possible values of the missing digits by applying the divisibility rule of 9. According to the rule, the sum of the digits of the number should be divisible by 9.
Let's analyze each number separately:
For the first number, 396 *5, the sum of the digits is 3 + 9 + 6 + 5 = 23. To make the number divisible by 9, the missing digit must be 6, resulting in the number 39665.
For the second number, 48675 *, the sum of the digits is 4 + 8 + 6 + 7 + 5 =30. The missing digit must be 6 to make the number divisible by 9, resulting in the number 486756.
For the third number, 349 *, the sum of the digits is 3 + 4 + 9 = 16. There are no possible values for the missing digit to make the number divisible by 9.
Therefore, the possible values for the missing digits in the given numbers are 6 and 6.