The numbers from 5 to 26 are written side-by-side as follows 567..2526. What is the remainder when resulting number gets divided by 9?
The Correct Answer and Explanation is:
To find the remainder when the large number formed by writing the numbers from 5 to 26 side-by-side (567…2526) is divided by 9, we can use a property of divisibility by 9: a number is divisible by 9 if the sum of its digits is divisible by 9. Therefore, instead of directly calculating the number, we can find the sum of all the digits and then determine the remainder when that sum is divided by 9.
Step 1: Determine the Sum of Digits
We need to sum the digits of all numbers from 5 to 26. The numbers from 5 to 26 are:
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26
Let’s sum the digits of each number:
- 5 → sum of digits = 5
- 6 → sum of digits = 6
- 7 → sum of digits = 7
- 8 → sum of digits = 8
- 9 → sum of digits = 9
- 10 → sum of digits = 1 + 0 = 1
- 11 → sum of digits = 1 + 1 = 2
- 12 → sum of digits = 1 + 2 = 3
- 13 → sum of digits = 1 + 3 = 4
- 14 → sum of digits = 1 + 4 = 5
- 15 → sum of digits = 1 + 5 = 6
- 16 → sum of digits = 1 + 6 = 7
- 17 → sum of digits = 1 + 7 = 8
- 18 → sum of digits = 1 + 8 = 9
- 19 → sum of digits = 1 + 9 = 10
- 20 → sum of digits = 2 + 0 = 2
- 21 → sum of digits = 2 + 1 = 3
- 22 → sum of digits = 2 + 2 = 4
- 23 → sum of digits = 2 + 3 = 5
- 24 → sum of digits = 2 + 4 = 6
- 25 → sum of digits = 2 + 5 = 7
- 26 → sum of digits = 2 + 6 = 8
Step 2: Calculate the Total Sum of Digits
Now, let’s add up all the digit sums: 5+6+7+8+9+1+2+3+4+5+6+7+8+9+10+2+3+4+5+6+7+85 + 6 + 7 + 8 + 9 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 2 + 3 + 4 + 5 + 6 + 7 + 85+6+7+8+9+1+2+3+4+5+6+7+8+9+10+2+3+4+5+6+7+8
Summing these gives: 5+6=11,11+7=18,18+8=26,26+9=35,35+1=36,36+2=38,38+3=41,41+4=455 + 6 = 11,\quad 11 + 7 = 18,\quad 18 + 8 = 26,\quad 26 + 9 = 35,\quad 35 + 1 = 36,\quad 36 + 2 = 38,\quad 38 + 3 = 41,\quad 41 + 4 = 455+6=11,11+7=18,18+8=26,26+9=35,35+1=36,36+2=38,38+3=41,41+4=45 45+5=50,50+6=56,56+7=63,63+8=71,71+9=80,80+10=90,90+2=9245 + 5 = 50,\quad 50 + 6 = 56,\quad 56 + 7 = 63,\quad 63 + 8 = 71,\quad 71 + 9 = 80,\quad 80 + 10 = 90,\quad 90 + 2 = 9245+5=50,50+6=56,56+7=63,63+8=71,71+9=80,80+10=90,90+2=92 92+3=95,95+4=99,99+5=104,104+6=110,110+7=117,117+8=12592 + 3 = 95,\quad 95 + 4 = 99,\quad 99 + 5 = 104,\quad 104 + 6 = 110,\quad 110 + 7 = 117,\quad 117 + 8 = 12592+3=95,95+4=99,99+5=104,104+6=110,110+7=117,117+8=125
The total sum of all the digits is 125.
Step 3: Find the Remainder When Divided by 9
Now, we divide the total sum of the digits by 9: 125÷9=13 remainder 8125 \div 9 = 13 \text{ remainder } 8125÷9=13 remainder 8
Thus, the remainder when the large number (formed by writing the numbers 5 to 26 side by side) is divided by 9 is 8.
Explanation
This approach is based on the fact that divisibility by 9 is determined by the sum of the digits of a number. By calculating the sum of the digits for all numbers from 5 to 26, we then determine the remainder when that sum is divided by 9. Instead of dealing with a very large number directly, we simplify the problem using properties of numbers and divisibility rules.
