Context: Let us consider the number that arose when dividing by . Multiply by numbers from to . Q. What are the products? What do you notice? Multiply by . What do you observe?

Another Peek Beyond the Point | IT

Question 14

Context: Let us consider the number $142857$ that arose when dividing $1$ by $7$ . Multiply $142857$ by numbers from $1$ to $6$ .

Q. What are the products? What do you notice?

Multiply $142857$ by $7$ . What do you observe?

Solution

Understand the Question

The number $142857$ is the repeating sequence of digits from the decimal expansion of $\dfrac{1}{7} = 0.\overline{142857}$ .
When multiplied by integers from $1$ to $6$ , $142857$ produces products with the exact same digits in the same cyclic order.
When multiplied by $7$ , the product results in a string of $9$ s because $\dfrac{1}{7} \times 7 = 1$ , which relates to $0.\overline{999999} = 1$ .

(i) Multiply $142857$ by numbers from $1$ to $6$ . What are the products? What do you notice?

Step 1 · Calculate Products from 1 to 6

Calculating the products for multipliers $1$ through $6$

\begin{aligned} 142857 \times 1 &= 142857 \\[0.4em] 142857 \times 2 &= 285714 \\[0.4em] 142857 \times 3 &= 428571 \\[0.4em] 142857 \times 4 &= 571428 \\[0.4em] 142857 \times 5 &= 714285 \\[0.4em] 142857 \times 6 &= 857142 \end{aligned}

Observation: Every product consists of the exact same six digits ( $1, 4, 2, 8, 5, 7$ ) in a cyclic shift (cyclic permutation).

Answer

(i) The products are $142857$ , $285714$ , $428571$ , $571428$ , $714285$ , and $857142$ . All products contain the same cyclic permutation of digits ( $1, 4, 2, 8, 5, 7$ ).

(ii) Multiply $142857$ by $7$ . What do you observe?

Step 1 · Calculate Product by 7

Multiplying by $7$

$142857 \times 7 = 999999$

Observation: The result is a six-digit number composed entirely of nines ( $999999$ ).

Answer

(ii) $999999$

Common Mistakes

Overlooking the Cyclic Order: Stating only that the digits are rearranged without noticing they maintain the same circular sequence ( $1 \to 4 \to 2 \to 8 \to 5 \to 7 \to 1$ ).
Multiplication Errors: Misaligning carries during standard vertical multiplication of 6-digit numbers by single-digit multipliers.

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Q14

Context: Let us consider the number $142857$ that arose when dividing $1$ by $7$ . Multiply $142857$ by numbers from $1$ to $6$ .

Q. What are the products? What do you notice?

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