1234567….99 is divide by 99 than reminder then find reminder is
The Correct Answer and Explanation is:
To find the remainder when dividing a number that consists of digits from 1 to 99 (i.e., the number 1234567…99) by 99, we can break this down into simpler steps. The number is a large sequence, but there is a helpful property we can use: the remainder of a number when divided by 99 depends only on the sum of its digits.
Steps:
- Sum the digits of the number:
The number consists of all integers from 1 to 99 written in sequence, so the digits involved are from 1 to 99. To find the sum of all the digits in the number, we can break it into parts:- The digits from 1 to 9 sum to 1+2+3+4+5+6+7+8+9=451 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 451+2+3+4+5+6+7+8+9=45.
- The digits from 10 to 99 require summing each of their tens and ones digits.
1+2+3+4+5+6+7+8+9=451 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 451+2+3+4+5+6+7+8+9=45.
These tens digits appear 10 times each (i.e., 10, 20, …, 90), so the total for the tens digits is 45×10=45045 \times 10 = 45045×10=450. For the ones digits, each appears exactly 9 times, so their sum is:
(1+2+3+…+9)×9=45×9=405(1 + 2 + 3 + … + 9) \times 9 = 45 \times 9 = 405(1+2+3+…+9)×9=45×9=405. Therefore, the total sum of all digits is:
45+450+405=90045 + 450 + 405 = 90045+450+405=900. - Find the remainder when the sum is divided by 99:
Now, we simply divide the sum of the digits (900) by 99 to find the remainder:
900÷99=9 remainder 9900 \div 99 = 9 \text{ remainder } 9900÷99=9 remainder 9.
Thus, the remainder when the number 123456789101112…99 is divided by 99 is 9.
