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Smith Number Calculator
Input a number to check if it's a Smith number, or specify a range to generate all Smith numbers within that range.
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What is a Smith Number?
A Smith Number is a special type of composite number defined as follows: It is a natural number (greater than 1 and not prime) where the sum of its digits equals the sum of the digits of its prime factors (considering the prime factors' repetition).
For example, 4 is a Smith number because:
Prime factorization : 4 = 2 × 2.Sum of prime factor digits: 2 + 2 = 4.
Sum of the digits of 4 is 4.
How to Determine if a Number is a Smith Number
Check if the number is composite : Ensure the number is not prime and is greater than 1. Perform prime factorization : Break the number into its prime factors .Calculate the sums :
Find the sum of the digits of the original number.
Find the sum of the digits of all its prime factors (including repeated factors).
Compare the sums : If the two sums are equal, the number is a Smith number.
Examples
Example 1: Is 22 a Smith Number?
Solution:
1. Check composite status:
22 is a composite number.
2. Prime factorization:
22 = 2 × 11
3. Sum of digits:
2 + 2 = 4
4. Sum of prime factor digits:
2 + 1 + 1 = 4
Result:
The sums are equal, so 22 is a Smith number.
Example 2: Is 28 a Smith Number?
Solution:
1. Check composite status:
28 is a composite number.
2. Prime factorization:
28 = 2 × 2 × 7
3. Sum of digits:
2 + 8 = 10
4. Sum of prime factor digits:
2 + 2 + 7 = 11
Result:
The sums are not equal, so 28 is not a Smith number.
Example 3: Is 2038 a Smith Number?
Solution:
1. Check composite status:
2038 is a composite number.
2. Prime factorization:
2038 = 2 × 1019
3. Sum of digits:
2 + 0 + 3 + 8 = 13
4. Sum of prime factor digits:
2 + 1 + 0 + 1 + 9 = 13
Result:
The sums are equal, so 2038 is a Smith number.
First 100 Smith Numbers
4
22
27
58
85
94
121
166
202
265
274
319
346
355
378
382
391
438
454
483
517
526
535
562
576
588
627
634
636
645
648
654
663
666
690
706
728
729
762
778
825
852
861
895
913
915
922
958
985
1086
1111
1165
1219
1255
1282
1284
1376
1449
1507
1581
1626
1633
1642
1678
1736
1755
1776
1795
1822
1842
1858
1872
1881
1894
1903
1908
1921
1935
1952
1962
1966
2038
2067
2079
2155
2173
2182
2218
2227
2265
2286
2326
2362
2366
2373
2409
2434
2461
2475
2484
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