Is the integer number 9,906 divisible by 9?

Is 9,906 divisible by 9?

Approach 1. Dividing numbers:

One integer A is divisible by another integer B, if after dividing them, A ÷ B, the remainder is zero.


9,906 is divisible by 9, if there is an integer 'n' such that:
9,906 = 'n' × 9.


Notice that dividing our numbers leaves a remainder:


9,906 ÷ 9 = 1,100 + 6;


There is no integer 'n' such that 9,906 = 'n' × 9.


9,906 is not divisible by 9.


Note:

1) If you subtract the remainder of the above operation, 6, from the initial number, 9,906, you'll get as a result a number that is evenly divisible by the second number, 9:


9,906 - 6 = 9,900;


9,900 = 1,100 × 9.


2) If you subtract the remainder of the above operation, 6, from the second number, 9, and then add the result to the initial number, 9,906, you'll get as a final result a number that is 'evenly divisible' by the second number, 9:

9 - 6 = 3;


9,906 + 3 = 9,909;


9,909 = 1,101 × 9.


9,906 is not divisible by 9
Dividing our numbers leaves a remainder.

Approach 2. Integer numbers prime factorization:

When are two numbers divisible?

Number 9,906 is divisible by number 9 if it has as factors all the prime numbers that occur in the prime factorization of 9.


Integer numbers prime factorization:

Prime Factorization of a number: finding the prime numbers that multiply together to make that number.


9,906 = 2 × 3 × 13 × 127;
9,906 is not a prime, is a composite number;


9 = 32;
9 is not a prime, is a composite number;



* Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself.
* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.


9,906 does not have (all) the prime factors of the number 9;


9,906 is not divisible by 9.


9,906 is not divisible by 9.

Final answer:
9,906 is not divisible by 9.
Dividing our numbers leaves a remainder.
9,906 does not have (all) the prime factors of the number 9.
Note:
9,900 is divisible by 9
9,909 is divisible by 9

More operations of this kind:

Online calculator: numbers' divisibility check

The latest numbers that were checked to see whether they are divisible or not

Is number 9,906 divisible by number 9? May 18 14:23 UTC (GMT)
Is number 5,133 divisible by number 718? May 18 14:23 UTC (GMT)
Is number 210 divisible by number 44? May 18 14:23 UTC (GMT)
Is number 2,607 divisible by number 762? May 18 14:23 UTC (GMT)
Is number 155,639 divisible by number 55,636? May 18 14:23 UTC (GMT)
Is number 4,972 divisible by number 2,486? May 18 14:23 UTC (GMT)
Is number 120,000 divisible by number 3? May 18 14:23 UTC (GMT)
Is number 2,282 divisible by number 847? May 18 14:23 UTC (GMT)
Is number 102,213 divisible by number 34,071? May 18 14:23 UTC (GMT)
Is number 358 divisible by number 130? May 18 14:23 UTC (GMT)
Is number 1,000,003 divisible by number 492,033? May 18 14:23 UTC (GMT)
Is number 196,655 divisible by number 39,331? May 18 14:23 UTC (GMT)
Is number 61,899 divisible by number 2? May 18 14:23 UTC (GMT)
integer numbers divisibility, see more...

Tutoring: What is the integer numbers divisibility? Divisibility rules.

Divisibility of integer numbers

Let's divide two different numbers, 12 and 15, by 4. When we divide 12 by 4, the quotient is 3 and the operation leaves no remainder. But when we divide 15 by 4, the quotient is 3 and the operation leaves a remainder of 3. We say that the number 12 is divisible by 4 and 15 is not. We also say that 4 is a divisor of 12, but is not a divisor of 15.

In general, we say that "a" is divisible by "b", if there is an integer number "n", so that: a = n × b. Number "b" is called the divisor of "a" ("n" is also a divisor of "a").

0 is divisible by any number other than zero itself.

Any number "a", different of zero, is divisible at least by 1 and itself, which are called improper divisors.

Some divisibility rules

The number 84 is divisible by 4 and 3 and is also divisible by 4 × 3 = 12. This is not true unless the two divisors are coprime.

In general, if "a" is divisible by both "m" and "n" and greatest common factor (m; n) = 1 (coprime numbers), then "a" it is also divisible by their product, (m × n).

Calculating divisors (factors) is very useful when simplifying fractions (reducing fractions to lower terms). The established rules for finding factors (divisors) are based on the fact that the numbers are written in the decimal system. Mutiples of 10 are divisible by 2 and 5, because 10 is divisible by 2 and 5; multiples of 100 are divisible by 4 and 25, because 100 is divisible by 4 and 25; multiples of 1000 are divisible by 8, because 1000 is divisible with 8. All the powers of 10, when divided by 3, or 9, have a remainder equal to 1.

Due to the rules of operation with remainders, we have the following remainders when dividing numbers by 3 or 9: 600 leaves a remainder equal to 6 = 1 × 6; 240 = 2 × 100 + 4 × 10, then the remainder will be equal to 2 × 1 + 4 × 1 = 6. On dividing a number by 3 or 9 the remainder will be equal to that left from dividing the sum of digits of that number by 3 or 9; 7,309 has the sum of the numbers 7 + 3 + 0 + 9 = 19, which is divided without a remainder to neither 3 nor 9. So 7,309 is not divisible by 3 or 9.

All even powers of 10, 100, 10,000, 1,000,000, etc., when divided by 11 left a remainder of 1, and the odd powers of 10, when divided by 11 left a remainder equal to 10 or 10 - 11 = -1. In this case, the alternating sum of the digits bears the same remainder as when dividing by 11, as if the whole number were being divided by 11. How to calculate the alternating sum is shown in the example below.

For instance, for the number: 85,976: 8 + 9 + 6 = 23, 5 + 7 = 12, the alternating sum of the digits: 23 - 12 = 11. So 85,976 is divisible by 11.

A number is divisible by:
  • 2 if the last digit is divisible by 2
  • 4 if the last two digits form a number divisible by 4;
  • 8, if the last three digits form a number divisible by 8;
  • 5 if the last digit is divisible by 5 (5 and 0)
  • 25, if the last two digits form a number divisible by 25
  • 3, if the sum of digits is divisible by 3;
  • 9, if the sum of digits is divisible by 9;
  • 11 if the alternating sum of digits is divisible by 11.

What is a prime number?

What is a composite number?

Prime numbers up to 1,000

Prime numbers up to 10,000

Sieve of Eratosthenes

Euclid's algorithm

Simplifying ordinary (common) math fractions (reducing to lower terms): steps to follow and examples