Balance Prime Exponents for Squares, Coprime Factors, and Equal Products
质因数指数的配平：完全平方、互质合数与等积分组
Prime exponents work like an inventory. Pair every prime copy for a square, keep complete prime-power packets apart for coprime factors, and match exponent totals when two products must be equal.
Read an exponent ledger
A positive whole number is a perfect square when it equals a whole number multiplied by itself. For a number greater than 1, this happens exactly when every prime exponent is even. Change the exponents and watch the square status update.
Why even exponents?
A square multiplies the same factorization by itself:
Each prime copy has a matching partner. That makes every exponent even.
Live exponent ledger
Read the fixed starting ledger: 2³ × 3² × 5
Answer for this fixed expression, even if you have changed the sliders. Exponent zero means no copies of that prime; all zero exponents give 1 = 1². “Square root” here means the positive square root.
Need a hint? Start here
Pair equal prime copies. An unpaired copy means an odd exponent.
How can I check my reasoning?
Look for odd exponents, not odd prime bases. The odd exponents belong to bases 2 and 5. The number is not a square.
Complete 1,470 to a perfect square
This follows Guided Practice 3. Multiply by exactly the missing prime copies—no more and no less.
original problem
Find the least positive integer x such that 1,470x is a perfect square.
Select the prime copies to add
Press a prime to include one extra copy in the multiplier; press it again to remove that copy. Set these selections as well as entering your answers.
Least means “repair only the odd exponents”
Adding another 7 would turn the even exponent 2 into the odd exponent 3. It would break the square condition.
Complete the worked example practice
Need a hint? Start here
Add exactly one copy of each prime with an odd exponent.
How can I check my reasoning?
Select 2,3,5; leave 7 unselected. Least multiplier 30. 1470×30=44100=210²; positive square root 210.
Use a general least-square-multiplier laboratory
Enter a whole number from 2 through 1,000,000,000, collect its primes with odd exponents, and multiply one extra copy of each. If it is already a square, the least positive multiplier is 1.
Challenge: start with 180
Need a hint? Start here
Factor 180 first; repair only the unpaired prime.
How can I check my reasoning?
180=2²×3²×5 has only one unpaired prime copy. Multiply by 5 to get 900; positive square root 30.
Complete 1,080 to a perfect square
This follows original Exercise 9. The same odd-exponent repair rule works again.
original exercise
Find the least possible value of a.
Repair ledger
Enter the worked example result
Need a hint? Start here
Distinguish the multiplier, the resulting square and its square root.
How can I check my reasoning?
Add one copy of each of 2,3,5. Multiplier 30; positive square root 180; square 32400.
Recover two coprime composite factors from LCM 126
This follows Worked example 3. “Composite” and “coprime” are separate conditions, and both must be checked.
For 4 and 9, the multiples begin 4, 8, 12, …, 36 and 9, 18, 27, 36. Their first shared multiple is 36. Their prime inventories are 2² and 3²: no prime copies overlap. A common multiple needs both complete inventories, so the smallest is 2² × 3² = 4 × 9. This is why coprime numbers have LCM equal to their product.
Three conditions
- Both numbers are composite.
- Their greatest common divisor is 1.
- Their least common multiple is 126.
Prime-power packets
Packet 9 can stand alone as a composite factor. Packets 2 and 7 must join to make the other composite factor.
Complete factor-pair audit
Enter the unique pair
Need a hint? Start here
Reject prime factors and pairs with a common divisor greater than 1.
How can I check my reasoning?
Reject pairs containing a prime or sharing a prime. Only 9 and 14 are both composite and coprime. Their sum is 23.
Keep prime-power packets together
If two factors are coprime, every full power of one prime must stay in only one factor. Splitting a packet would create a shared prime.
Why a packet cannot split
Suppose one copy of 7 goes into each factor. Then both factors are divisible by 7:
That contradicts “coprime.” Therefore all copies of 7 in the fixed product must stay together.
What the laboratory checks
For each factor pair it verifies:
- both factors are composite;
- their GCD is 1;
- their LCM equals the target.
Coprime-composite factor laboratory
Challenge: target LCM 360
For each valid pair, put its smaller member in the smaller-factor list and its larger member in the larger-factor list. These are two separate lists; their positions do not pair entries together.
Need a hint? Start here
A full prime-power packet must stay together if the factors are coprime.
How can I check my reasoning?
Keep only pairs in which both factors are composite. Two pairs: (8,45) and (9,40). Smaller-factor list: 8,9; larger-factor list: 40,45.
Enumerate every pair in original Exercise 8
The worked example asks for two positive whole numbers divisible by 30 whose product is 308 × 450. The printed conditions allow four unordered pairs, not one.
original exercise
Both A and B are divisible by 30.
Remove the required factor 30 from both
Factor pairs of 154 and the original pairs
Record the complete result
Need a hint? Start here
Write the numbers as 30x and 30y; divide the product by 900.
How can I check my reasoning?
xy=154. Its unordered factor pairs are (1,154),(2,77),(7,22),(11,14). Four original pairs: 30×4620; 60×2310; 210×660; 330×420.
Split eight numbers into two equal-product groups
This follows original Exercise 11. Use every card once, with exactly four cards in each group. Swapping Group A and Group B counts as the same unordered split. Build a valid split before checking Mission 8. Click each card to cycle through unassigned, Group A, and Group B. The validator accepts every mathematically valid split.
original number cards
A group with no cards displays product 1, the starting value for multiplication. It is not a completed group.
Split these eight numbers into two groups whose products are equal:
Group A
Group B
Why the total product must be a square
If both group products equal T, then:
The total prime exponents are:
So each group must have half of every exponent.
Compare after your attempt
The target product for each group is:
State what the full search proves
Need a hint? Start here
Use four cards in each group and balance every prime exponent.
How can I check my reasoning?
Swapping group labels does not give a new partition. Two unordered splits: (14, 33, 75, 169) versus [35, 30, 39, 143], and (14, 39, 75, 143) versus [33, 35, 30, 169]. Each group product is 5855850.
Prime-exponent workshop
Correct all eight questions to complete the workshop.
Need a hint? Start here
Choose the strategy first: factor, apply the conditions, and multiply back. Revisit the relevant local hint above.
How can I check my reasoning?
For equal-product groups, halve the total prime exponents. Answers: all even; 30; 5; 23; 2; 4; 5855850; 2.
Exit ticket
Complete all ten missions, including 5 out of 5 on this exit ticket, to earn the certificate.
Certificate of completion
Prime-Exponent Balance Architect
Awarded to Student for balancing exponents, preserving coprimality, and proving equal-product partitions.
Need a hint? Start here
Work without the laboratories first. Explain which prime copies or factor pairs your answer uses, then verify the result.
How can I check my reasoning?
Keep the multiplier distinct from the resulting square root. Answers: 7; 210; 9; 4; 2.