Name: Prime Numbers From 1 to 100
Uploaded: 2019-07-19
Duration: 7 min 49 s
Description: The number 2 is the only even prime number. But, how do you find the prime numbers between 1 and 100? This video explains the easiest way to find prime numbers.

Question

All the

2

-digit numbers from

19

to

93

are written consecutively to form the number

N = 19202122 \dots 919293

. Find the largest power of

3

that divides

N

.

Accepted Answer

A number is divisible by $3$ if the sum of its digits is divisible by $3$ and it is divisible by $9$ if the sum of digits is divisible by $9 .$

To test the divisibility of the number $N = 19202122 \dots 919293$ by $3$ or $9$ we should find the sum of the digits of $N .$

The sum of the natural numbers $1 + 2 + 3 + . . . + n = \frac{n (n + 1)}{2} .$

The sum of the unit's digit of the numbers $20, 21, 22, . . ., 29$ is $0 + 1 + 2 + . . . + 9 = \frac{9 (9 + 1)}{2} = 45$ and the sum of the ten's digit of these numbers is $10 \times 2 = 20$ thus the sum of digits of all these numbers is $45 + 20 = 65 .$

Similarly, the sum of the digits of the numbers $30, 31, 32, . . ., 39$ is $45 + 30 = 75 .$

And, the sum of the digits of the numbers $40, 41, 42, . . ., 49$ is $45 + 40 = 85 .$

And, the sum of the digits of the numbers $50, 51, 52, . . ., 59$ is $45 + 50 = 95 .$

And, the sum of the digits of the numbers $60, 61, 62, . . ., 69$ is $45 + 60 = 105 .$

And, the sum of the digits of the numbers $70, 71, 72, . . ., 79$ is $45 + 70 = 115 .$

And, the sum of the digits of the numbers $80, 81, 82, . . ., 89$ is $45 + 80 = 125 .$

Hence, the sum of digits of all the numbers from $19$ to $93$ is $1 + 9 + 65 + 75 + 85 + 95 + 105 + 115 + 125 + 9 + 0 + 9 + 1 + 9 + 2 + 9 + 3 = 717 .$

We find the sum of the digits of $N$ i.e. $717$ is divisible by $3$ but not by $9 .$

Thus, the highest power of $3$ dividing $N$ is $1 .$

All the $2$ -digit numbers from $19$ to $93$ are written consecutively to form the number $N = 19202122 \dots 919293$ . Find the largest power of $3$ that divides $N$ .

Important Questions on Number Theory

Learn and Prepare for any exam you want !

All the 2-digit numbers from 19 to 93 are written consecutively to form the number N=19202122…919293. Find the largest power of 3 that divides N.

Important Questions on Number Theory

Learn and Prepare for any exam you want !

All the $2$ -digit numbers from $19$ to $93$ are written consecutively to form the number $N = 19202122 \dots 919293$ . Find the largest power of $3$ that divides $N$ .