Smith Number Checker

Smith Number Checker is a free online tool used to check if a number is a Smith number.

Smith Number Checker

Eg, Is 202 a Smith number? Enter 202 into the input box and click Calculate button.

What is a Smith number?

Smith number is defined as a composite number for a given number base, where the sum of the digits of the number is equal to the sum of the digits in its prime factorization for the given number base.

For example, 94 is a Smith number.

The sum of the digits of 94 is 9 + 4 = 13.

The prime factors of 94 are 2, 47. The sum of the digits of the prime factors is 2 + 4 + 7 = 13.

All prime numbers are Smith numbers because they satisfy this condition. Note that 1 is not a prime number, so 1 is not a Smith number.

For example, the prime number 13.

The sum of digits of 13 is 1 + 3 = 4.

Because 13 is a prime number, so the prime factor of 13 is also 13. No doubt, 13 is a Smith number.

How to check if a number is a Smith number

Obviously, to determine whether a number is a Smith number, at least the following steps must be performed:

1Calculate the sum of the digits of the number.
2Calculate prime factorization of the number.
3Calculate the sum of the digits in its prime factors.
4Judge whether the values of step 1 and step 3 are equal.

For example, Is 121 a Smith number?

First, calculate the sum of the digits of 121

1 + 2 + 1 = 4

Second, calculate the prime factorization of 121. With the prime factorization calculator, we can easily get the prime factorization of 121 is

121 = 11 * 11

Third, calculate the sum of the digits in the prime factors of 121. Through the second step, we know that the prime factor of 121 is two 11. So, the sum of their digits is

1 + 1 + 1 + 1 = 4

Consistent with the result calculated in the first step, so, 121 is a Smith number.

First 100 Smith numbers

2	3	4	5	7	11	13	17	19	22
23	27	29	31	37	41	43	47	53	58
59	61	67	71	73	79	83	85	89	94
97	101	103	107	109	113	121	127	131	137
139	149	151	157	163	166	167	173	179	181
191	193	197	199	202	211	223	227	229	233
239	241	251	257	263	265	269	271	274	277
281	283	293	307	311	313	317	319	331	337
346	347	349	353	355	359	367	373	378	379
382	383	389	391	397	401	409	419	421	431

How to use the Smith number checker

The procedure to use the Smith number checker is as follows:

1Enter the positive number.
2Click Calculate button to verify whether the input number is a Smith number.
3Click the Reset button to start a new verification.

Solved examples using the Smith number checker

Example 1: Is 357 a Smith number?

Enter 357 into the input box and click Calculate button, as shown in the figure, 357 is not a Smith number.

The sum of the digits of 357 is

3 + 5 + 7 = 15

The prime factors of 357 are 3, 7 and 17. The sum of their digits is

3 + 7 + 1 + 7 = 18

Example 2: Is 2038 a Smith number?

Enter 2038 into the input box and click Calculate button, as shown in the figure, 2038 is a Smith number.

The sum of the digits of 2038 is

2 + 0 + 3 + 8 = 13

The prime factors of 2038 are 2 and 1019. The sum of their digits is

2 + 1 + 0 + 1 + 9 = 13

Smith Number Checker