Every natural number (greater than 1) is divisible by two natural numbers, i.e. 1 and itself. Therefore, every natural number greater than 1 has at least two factors. Prime numbers have only two factors.
Note: 1 is neither a prime nor a composite number.
Divisibility hints:
2- Any natural number is divisible by 2 when the the digit at its unit place is any one among 0,2,4,6,8. Example: 22,4,36.
3- A natural number is divisible by 3 when the sum of its digits is divisible by 3.
For example, To check if 35892 is divisible by 3, we find the sum of digits.
3+5+8+9+2=27
27 is divisible by 3. Therefore, 35892 is divisible by 3.
4- A natural number is divisible by 4 when the number formed by digits at ten's and unit's place respectively is divisible by 4.
For example, To check if 35892 is divisible by 4:
The number formed by ten's and unit's place is 92
92 is divisible by 4. Therefore, 35892 is divisible by 4.
5- Any natural number is divisible by 5 when the the digit at its unit place is any one among 0 or 5. Example: 25, 93670.
7- A natural number is divisible by 7 when the difference between twice the digit at unit's place and the number formed by rest of the digits is 0 or a multiple of 7.
For example, Consider 196.
Twice the digit at unit's place=2*6=12
Number formed by rest of the digits=19
19-12=7
Therefore, 196 is divisible by 7.
11- Take the sum S1 of the digits at odd places and the sum S2 of the digits at even places. If the difference between S1 and S2 is 0 or any multiple of 11, the number is divisible by 11.
For example, To check if 3852101 is divisible by 11:
S1= 1+1+5+3=10, S2= 0+2+8=10, S1-S2=0
Therefore, the given number 3852101 is divisible by 11.
2^n- If the number formed by last n digits of the natural number is divisible by 2^n, the number is divisible by 2^n.
For example,
If the number formed by last 4 digits of a natural number is divisible by 16, the natural number is divisible by 16. Consider 7856981600.
Similarly, If the number formed by last 5 digits of a natural number is divisible by 32, the natural number is divisible by 32.
5^n- If the number formed by last n digits of the natural number is divisible by 5^n, the number is divisible by 5^n.
For example,
If the number formed by last 2 digits of a natural number is divisible by 25, the natural number is divisible by 25. Consider 625.
3^n- If the sum of the digits of a natural number is divisible by 3^n, the natural number is divisible by 3^n.
For example,
If the sum of the digits of a natural number is divisible by 9, the natural number is divisible by 9. Consider 728109.
6^n- If a natural number is divisible by 2^n and 3^n, it is divisible by 6^n.
For example,
If a natural number is divisible by 4 and 9, it is divisible by 36.
10^n- If a natural number has n consecutive zeroes beginning from the unit's place of the number, it is divisible by 10^n.
For example,
If a number has 3 consecutive zeroes beginning from the unit's place, it is divisible by 1000. Consider 378000.
Note: If a natural number is divisible by two co-prime numbers, it is also divisible by the product of the co-prime numbers.
Thus, if a natural number is divisible by 3 and 4, it is also divisible by 12. Consider 576.
The vice versa of the statement is also true.
Thus to check if a number is divisible by 42, it suffices to check if it is divisible by 6 and 7.
- Rishabh Bidya
Note: 1 is neither a prime nor a composite number.
Divisibility hints:
2- Any natural number is divisible by 2 when the the digit at its unit place is any one among 0,2,4,6,8. Example: 22,4,36.
3- A natural number is divisible by 3 when the sum of its digits is divisible by 3.
For example, To check if 35892 is divisible by 3, we find the sum of digits.
3+5+8+9+2=27
27 is divisible by 3. Therefore, 35892 is divisible by 3.
4- A natural number is divisible by 4 when the number formed by digits at ten's and unit's place respectively is divisible by 4.
For example, To check if 35892 is divisible by 4:
The number formed by ten's and unit's place is 92
92 is divisible by 4. Therefore, 35892 is divisible by 4.
5- Any natural number is divisible by 5 when the the digit at its unit place is any one among 0 or 5. Example: 25, 93670.
7- A natural number is divisible by 7 when the difference between twice the digit at unit's place and the number formed by rest of the digits is 0 or a multiple of 7.
For example, Consider 196.
Twice the digit at unit's place=2*6=12
Number formed by rest of the digits=19
19-12=7
Therefore, 196 is divisible by 7.
11- Take the sum S1 of the digits at odd places and the sum S2 of the digits at even places. If the difference between S1 and S2 is 0 or any multiple of 11, the number is divisible by 11.
For example, To check if 3852101 is divisible by 11:
S1= 1+1+5+3=10, S2= 0+2+8=10, S1-S2=0
Therefore, the given number 3852101 is divisible by 11.
2^n- If the number formed by last n digits of the natural number is divisible by 2^n, the number is divisible by 2^n.
For example,
If the number formed by last 4 digits of a natural number is divisible by 16, the natural number is divisible by 16. Consider 7856981600.
Similarly, If the number formed by last 5 digits of a natural number is divisible by 32, the natural number is divisible by 32.
5^n- If the number formed by last n digits of the natural number is divisible by 5^n, the number is divisible by 5^n.
For example,
If the number formed by last 2 digits of a natural number is divisible by 25, the natural number is divisible by 25. Consider 625.
3^n- If the sum of the digits of a natural number is divisible by 3^n, the natural number is divisible by 3^n.
For example,
If the sum of the digits of a natural number is divisible by 9, the natural number is divisible by 9. Consider 728109.
6^n- If a natural number is divisible by 2^n and 3^n, it is divisible by 6^n.
For example,
If a natural number is divisible by 4 and 9, it is divisible by 36.
10^n- If a natural number has n consecutive zeroes beginning from the unit's place of the number, it is divisible by 10^n.
For example,
If a number has 3 consecutive zeroes beginning from the unit's place, it is divisible by 1000. Consider 378000.
Note: If a natural number is divisible by two co-prime numbers, it is also divisible by the product of the co-prime numbers.
Thus, if a natural number is divisible by 3 and 4, it is also divisible by 12. Consider 576.
The vice versa of the statement is also true.
Thus to check if a number is divisible by 42, it suffices to check if it is divisible by 6 and 7.
- Rishabh Bidya
