Express each of the following odd numbers as the sum of 3 prime numbers: a. 29, b. 83, c. 55, d. 53, e. 99

20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.

A prime number has no positive divisors other than 1 and itself.

Check each number for divisibility by primes up to its square root.

$\sqrt{20} \approx 4.47$, test primes: 2, 3.
$20 \div 2 = 10$ (integer). Not prime.

$\sqrt{21} \approx 4.58$, test primes: 2, 3.
Sum of digits: $2+1=3$, divisible by 3.
$21 \div 3 = 7$ (integer). Not prime.

$\sqrt{22} \approx 4.69$, test primes: 2, 3.
$22 \div 2 = 11$ (integer). Not prime.

$\sqrt{23} \approx 4.80$, test primes: 2, 3.
Odd, not divisible by 2.
Sum of digits: $2+3=5$, not divisible by 3.
Prime.

Even and greater than 2.
$24 \div 2 = 12$ (integer). Not prime.

$\sqrt{25} = 5$, test primes: 2, 3, 5.
Ends in 5.
$25 \div 5 = 5$ (integer). Not prime.

Even and greater than 2.
$26 \div 2 = 13$ (integer). Not prime.

$\sqrt{27} \approx 5.20$, test primes: 2, 3, 5.
Sum of digits: $2+7=9$, divisible by 3.
$27 \div 3 = 9$ (integer). Not prime.

Even and greater than 2.
$28 \div 2 = 14$ (integer). Not prime.

$\sqrt{29} \approx 5.39$, test primes: 2, 3, 5.
Odd, not divisible by 2.
Sum of digits: $2+9=11$, not divisible by 3.
Does not end in 0 or 5, not divisible by 5.
Prime.

Even and ends in 0.
$30 \div 2 = 15$ (integer). Not prime.

The prime numbers are 23 and 29.