Determine the value of A if the ten digit number 512A51739A is
divisible by 7, 11 and 13.
To determine the value of A, we need to check if the ten-digit number 512A51739A is divisible by 7, 11, and 13.
First, let's start with divisibility by 7. One way to determine if a number is divisible by 7 is to use the rule of alternating sum. We need to subtract twice the units digit from the remaining number formed by the other digits. Let's calculate it step by step:
Step 1: Remove the units digit (A): 512A51739
Step 2: Multiply the units digit by 2 and subtract it from the remaining number: 512A5173 - 2A = 512A5173 - 20A = 512A5173 - 19A
Now we need to check if this new number, 512A5173 - 19A, is divisible by 7. If it is, then the original number is also divisible by 7.
Moving on to divisibility by 11. For a number to be divisible by 11, the difference between the sum of the odd-positioned digits and the sum of the even-positioned digits has to be either 0 or a multiple of 11. Let's calculate it step by step:
Step 1: Separate the digits into odd and even positions:
Odd positions: 5, A, 5, 7, 9 = 5+5+9 = 19
Even positions: 1, 2, A, 3, A = 1+2+3 = 6
Step 2: Calculate the difference: 19 - 6 = 13
Now we need to determine if this difference, 13, is divisible by 11. If it is, then the original number is also divisible by 11.
Lastly, for divisibility by 13, we will use a similar method as divisibility by 7 but with a different rule. To determine if a number is divisible by 13, multiply the units digit by 9 and subtract it from the remaining number:
Step 1: Remove the units digit (A): 512A51739
Step 2: Multiply the units digit by 9 and subtract it from the remaining number: 512A5173 - 9A = 512A5173 - 9A
Now we need to check if this new number, 512A5173 - 9A, is divisible by 13. If it is, then the original number is also divisible by 13.
By applying these divisibility rules for 7, 11, and 13, we can determine the value of A that satisfies all three conditions.