Below is a test for divisibility of a number \( n \) by 7 . Let \( r \) be the last digit of \( n \), and let \( q \) be the number obtained from \( n \) by dropping its last digit (that is, \( n=10 q+r \) ). Then \( n \) is divisible by 7 if, and only if, the number \( m=q-2 r \) is divisible by 7 . For example, \( n=40,061 \) is divisible by 7 if, and only if, \( 4006-2=4004 \) is divisible by 7 ; 4004 is divisible by 7 if, and only if, \( 400-8=392 \) is divisible by 7 ; and finally, 392 is divisible by 7 if, and only if, \( 39-4=35 \) is divisible by 7 . Since \( 35=7 \) • 5 , it follows that 40,061 is divisible by 7 . Use the test to check each of the following numbers for divisibility by 7 , without actually doing the division. \( \begin{array}{ll}\text { (a) } 497 & \text { (b) } 2811 \\ \text { (c) } 18,402\end{array} \) (a) Is 497 divisible by 7 ? Yes

To determine if the numbers \( 497 \), \( 2811 \), and \( 18402 \) are divisible by \( 7 \) using the provided test, we will follow the steps outlined in the problem statement. ### (a) Checking \( 497 \) 1. **Identify \( r \) and \( q \)**: - Last digit \( r = 7 \) - Remaining number \( q = 49 \) 2. **Calculate \( m \)**: \[ m = q - 2r = 49 - 2 \times 7 = 49 - 14 = 35 \] 3. **Check if \( m \) is divisible by \( 7 \)**: - \( 35 \div 7 = 5 \) (which is an integer) Since \( 35 \) is divisible by \( 7 \), it follows that \( 497 \) is also divisible by \( 7 \). ### (b) Checking \( 2811 \) 1. **Identify \( r \) and \( q \)**: - Last digit \( r = 1 \) - Remaining number \( q = 281 \) 2. **Calculate \( m \)**: \[ m = q - 2r = 281 - 2 \times 1 = 281 - 2 = 279 \] 3. **Check if \( m \) is divisible by \( 7 \)**: - Now we need to check \( 279 \): - Last digit \( r = 9 \) - Remaining number \( q = 27 \) \[ m = q - 2r = 27 - 2 \times 9 = 27 - 18 = 9 \] 4. **Check if \( 9 \) is divisible by \( 7 \)**: - \( 9 \div 7 \) is not an integer. Since \( 9 \) is not divisible by \( 7 \), it follows that \( 2811 \) is not divisible by \( 7 \). ### (c) Checking \( 18402 \) 1. **Identify \( r \) and \( q \)**: - Last digit \( r = 2 \) - Remaining number \( q = 1840 \) 2. **Calculate \( m \)**: \[ m = q - 2r = 1840 - 2 \times 2 = 1840 - 4 = 1836 \] 3. **Check if \( m \) is divisible by \( 7 \)**: - Now we need to check \( 1836 \): - Last digit \( r = 6 \) - Remaining number \( q = 183 \) \[ m = q - 2r = 183 - 2 \times 6 = 183 - 12 = 171 \] 4. **Check if \( 171 \) is divisible by \( 7 \)**: - Now we need to check \( 171 \): - Last digit \( r = 1 \) - Remaining number \( q = 17 \) \[ m = q - 2r = 17 - 2 \times 1 = 17 - 2 = 15 \] 5. **Check if \( 15 \) is divisible by \( 7 \)**: - \( 15 \div 7 \) is not an integer. Since \( 15 \) is not divisible by \( 7 \), it follows that \( 18402 \) is not divisible by \( 7 \). ### Summary of Results: - (a) \( 497 \) is divisible by \( 7 \): **Yes** - (b) \( 2811 \) is divisible by \( 7 \): **No** - (c) \( 18402 \) is divisible by \( 7 \): **No**