Use Parity and Remainders to Pin Down Primes
利用奇偶性与余数锁定质数
Use the fact that 2 is the only even prime, then organize numbers by their remainders to eliminate whole families of impossible cases before testing individual primes.
Use the one exceptional even prime
Every prime except 2 is odd. This single exception turns many long prime searches into short parity arguments.
The special prime
Every prime greater than 2 is odd.
Two-prime sums
So an odd sum of two primes forces one prime to be 2.
Several odd primes
Parity composer
Check the parity toolkit
Need a hint? Start here
Parity means whether a number is odd or even.
How to check my reasoning
An odd count of odd addends gives an odd sum; adding 2 does not change it. 2; odd; 2.
Worked example 1: seven consecutive primes
Seven consecutive primes are named a, b, c, d, e, f, g from greatest to least. Their total is even. Determine c.
Why the block must contain 2
Consecutive primes occupy neighboring positions in the prime list, with no prime skipped. They need not be consecutive whole numbers.
If the seven primes were all odd, their sum would be odd.
The total is even, so the block must include the only even prime, 2.
Because the primes are consecutive, the block is exactly the first seven primes.
Consecutive-prime block explorer
Move the slider until the only even-sum block appears.
Descending labels
The third entry from the greatest end is c.
Finish the worked example deduction
Need a hint? Start here
Seven odd primes would have an odd sum.
How to check my reasoning
The consecutive block must include 2, so it is the first seven primes. Read labels from greatest to least. 2; c=11; first seven primes.
Guided Practice 1: three distinct primes total 40
The word distinct means all three primes must be different. Parity first forces one prime; then a complete pair search handles the remaining total.
Force the even prime
Three odd primes would have an odd sum. Since 40 is even, one of the three primes must be:
The remaining two primes must total:
Search prime pairs totaling 38
Run the search to inspect every unordered prime pair.
Enter the three primes from least to greatest
Run the complete pair search before checking your answers.
Need a hint? Start here
All three primes must be different. Run the pair search.
How to check my reasoning
Parity forces 2. The remaining pair totals 38; 19+19 repeats a prime. 2,7,31 in increasing order.
Worked example 2: prove a prime sum is impossible
Can two primes add to the 20-digit number made entirely of 1s? The worked example combines parity with a digit-sum divisibility test.
The target number
This is a 20-digit odd number.
Reveal the proof one step at a time
Why the digit sum finishes the proof
Its digit sum is:
So it is divisible by 9 and is greater than 9. It is composite, not prime.
Audit the impossibility proof
Reveal all four proof steps before checking your answers.
Need a hint? Start here
Reveal all four steps. The odd sum forces one addend to be 2.
How to check my reasoning
Subtracting 2 changes the final 11 into 09; the digit sum is 27. No; forced prime 2; one-digit composite divisor 9.
Guided Practice 3: a weighted prime equation
Two primes p and q satisfy 3p + 2q = 2000. Use parity before doing any primality search.
Parity forces p
2q is even and 2000 is even, so 3p must be even. Since 3 is odd, p must be even.
Candidate explorer
Complete the equation
Need a hint? Start here
Both 2q and 2000 are even.
How to check my reasoning
Therefore 3p is even; since 3 is odd, p must be the only even prime. p=2; q=997; p+q=999.
Guided Practice 4: opposite cube faces
The sum of the two numbers in each pair of opposite faces is the same for all three pairs. The faces opposite 18, 14, and 35 contain primes a, b, and c.
Common-sum explorer
Parity shortcut
If the common sum were even, then a = S − 18 and b = S − 14 would both be even. The only even prime is 2, but the two equations cannot both work with the same even sum.
So the common sum is odd. Then:
Complete the cube
Run the common-sum search before checking. The parity argument proves uniqueness for all sums; the tool checks only the displayed range 20 through 100.
Need a hint? Start here
Opposite-pair sums are equal: 18+a=14+b=35+c.
How to check my reasoning
An even common sum would force both a and b to be 2, contradicting the different fixed faces. Thus the sum is odd and c=2. Common sum 37; c=2; a+b+c=19+23+2=44. Run the displayed search to complete the check.
Use remainder classes to pin down prime chains
Three expressions can cover all three possible remainders after division by 3. One must then be divisible by 3—and if it is prime, it must equal 3.
Prime-chain explorer
p, p + 10, p + 14
Because:
the three terms occupy remainder classes 0, 1, and 2 in some order. The divisible-by-3 term must itself be the prime 3.
17
p, p + 20, p + 40
Use the remainder-class argument
Run a search through p≤200 before checking. The remainder argument proves the result for every prime p; the search confirms cases only in its stated range.
Need a hint? Start here
Reduce the offsets modulo 3 and run the comparison search.
How to check my reasoning
One member is divisible by 3. The larger members exceed 3, so the first member must be 3. p=3 for both chains; the three terms occupy all three remainder classes.
Explore prime sums systematically
Parity predicts whether 2 is necessary. A complete search counts each group of prime terms once, in increasing order (or nondecreasing order if repeats are allowed). Rearrangements such as 2+7+31 and 31+2+7 count as the same representation.
Prime-sum enumerator
Two transfer conclusions
exerciseIf two primes total 2001, the odd sum forces one prime to be 2. The other is 1999, which is prime.
If primes satisfy x + y = z, then the two addends cannot both be odd, because their sum would be an even prime greater than 2. Therefore one addend is 2.
Transfer the method
Need a hint? Start here
For two primes with an odd sum, one must be 2.
How to check my reasoning
2001−2=1999. For x+y=z, two odd addends would make an even number greater than 2. 2; 1999; product 3998; smallest prime 2.
Parity-and-remainder workshop
Correct all eight questions to complete the workshop.
1. What is the only even prime?
2. The sum of seven odd primes is:
3. Seven consecutive primes a>b>c>d>e>f>g have an even total. What is c?
4. What is the largest of the three distinct primes totaling 40?
5. Can the 20-digit all-ones number be written as a sum of two primes?
6. If 3p + 2q = 2000 for primes p and q, what is p + q?
7. The primes a,b,c satisfy 18+a=14+b=35+c. What is a+b+c?
8. If p, p+20, and p+40 are all prime, what is p?
Need a hint? Start here
Use parity before testing large candidates.
How to check my reasoning
Check distinctness, descending labels, and which prime is forced to equal 2 or 3. 2; odd; 11; 31; no; 999; 44; 3.
Exit ticket
Complete Missions 1–8, earn 8/8 in the workshop, and earn 5/5 here to unlock your certificate. The optional reflection is not automatically graded.
1. If two primes have an odd sum, one of them must be:
2. Seven consecutive primes have an even total. Their block must begin with which smallest prime?
3. Three distinct primes total 40. What is the middle prime?
4. After subtracting 2 from the 20-digit all-ones number, what is the digit sum?
5. Two primes total 45. What is the larger prime?
Optional reflection — not automatically graded
Certificate of completion
Prime Parity & Remainder Navigator
has completed Lesson 17.2 and demonstrated accurate use of parity, divisibility, remainder classes, complete candidate searches, and prime verification.
Teaching notes
Need a hint? Start here
Every prime except 2 is odd.
How to check my reasoning
Explain your method in the reflection box. The checker grades the answer; your explanation is for you to review.