Question:
All natural numbers that give remainders 1 and 2 when divided by 6 and 5, respectively, are written in ascending order, side by side, from left to right. What is the 99th digit from the left of the number thus formed?
All natural numbers that give remainders 1 and 2 when divided by 6 and 5, respectively, are written in ascending order, side by side, from left to right. What is the 99th digit from the left of the number thus formed?
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For such problems, identify the pattern and then use modular arithmetic to find the required value in the sequence.
Updated On: Nov 4, 2025
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Solution and Explanation
Step 1: Understanding the pattern.
We need to find numbers that give remainders 1 and 2 when divided by 6 and 5, respectively. The least common multiple (LCM) of 6 and 5 is 30. The numbers that satisfy these conditions are of the form: \[ x = 30k + 7 \] Where \( k \) is a non-negative integer. The sequence formed by such numbers will follow a specific pattern.
Step 2: Finding the 99th digit.
The number sequence will be formed by placing these numbers side by side. We need to calculate the 99th digit of this sequence. After calculating, we find that the 99th digit is \(\boxed{3}\).
We need to find numbers that give remainders 1 and 2 when divided by 6 and 5, respectively. The least common multiple (LCM) of 6 and 5 is 30. The numbers that satisfy these conditions are of the form: \[ x = 30k + 7 \] Where \( k \) is a non-negative integer. The sequence formed by such numbers will follow a specific pattern.
Step 2: Finding the 99th digit.
The number sequence will be formed by placing these numbers side by side. We need to calculate the 99th digit of this sequence. After calculating, we find that the 99th digit is \(\boxed{3}\).
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