So many nine's!

Divide the no. which entirely consist nine's.

• 9/7 = 1.28
• 99/7 = 14.14
• 999/7 = 142.71
• 9999/7 = 1428.42
• 99999/7 = 14285.57
• 999999/7 = 142857

AS,THE ANSWER SHOULD BE IN INTEGERS+. THEREFORE,ANSWER WILL BE 142857.

For each digit, we need to find the formula, $(7D + P)$ % $10 = 9$ , with $D$ being the single-digit integer we are looking for and $P$ being the digit already in the Place we are trying to turn into a $9$ . Remember % means modulus.

For the $1$ 's place, the formula is $(7D + 0)$ % $10 = 9$ . $D$ is $7$ because $(7 \times 7 + 0)$ % $10 = (49 + 0)$ % 10 = 49 \% \(10 = 9

For the $10$ 's place, the formula is $(7D + 4)$ % $10 = 9$ . $D$ is $5$ because $(7 \times 5 + 4)$ % $10 = (35 + 4)$ % 10 = 39 \% \(10 = 9

For the $100$ 's place, the formula is $(7D + 3)$ % $10 = 9$ . $D$ is $8$ because $(7 \times 8 + 3)$ % $10 = (56 + 3)$ % 10 = 59 \% \(10 = 9

For the $1,000$ 's place, the formula is $(7D + 5)$ % $10 = 9$ . $D$ is $2$ because $(7 \times 2 + 5)$ % $10 = (14 + 5)$ % 10 = 19 \% \(10 = 9

For the $10,000$ 's place, the formula is $(7D + 1)$ % $10 = 9$ . $D$ is $4$ because $(7 \times 2 + 1)$ % $10 = (28 + 1)$ % 10 = 29 \% \(10 = 9

For the $100,000$ 's place, the formula is $(7D + 2)$ % $10 = 9$ . $D$ is $1$ because $(7 \times 1 + 2)$ % $10 = (7 + 2)$ % 10 = 9 \% \(10 = 9

Thus, the smallest integer is $142,857$ .