Read GCD and LCM from Prime Exponents
从质因数指数中读出最大公约数与最小公倍数
Align the prime factorizations, fill missing primes with exponent zero, take the smallest exponent for the GCD, and the largest exponent for the LCM.
Decide whether the story asks for a common divisor or a common multiple
Here all input numbers, divisors, and multiples are positive whole numbers. A divisor fits into a number with no remainder. A positive multiple is that number times 1, 2, 3, and so on. The words greatest and least positive tell us which common value to choose.
The greatest positive integer that divides every given number.
gcd(a,b,c)The worked example also writes this as (a,b,c).
The least positive integer divisible by every given number.
lcm(a,b,c)The worked example also writes this as [a,b,c].
Factor-and-multiple explorer
Checkpoint for the fixed numbers shown below
Need a hint? Start here
List shared factors for a greatest dividing unit; list shared multiples for a first repeat.
Check why it works
Shared divisors are 1,2,3,6; the first shared positive multiple is 36. GCD 6; LCM 36. A short displayed multiples list may end before the first common multiple.
Align every prime base—even when its exponent is zero
The worked example rewrites all three numbers with the same prime columns. A prime that is absent has exponent zero because p0=1.
original numbers
Prime 2
Prime 11
Checkpoint for the fixed numbers shown below
Need a hint? Start here
A missing prime contributes no copies: its exponent is zero. Compare one column at a time.
Check why it works
Then use all three original numbers for the larger checkpoint; include zeros. Prediction: minimum 1, maximum 2. Checkpoint: exponent of 2 in 819 is 0; exponent of 11 in 2520 is 0; minimum exponent of 3 is 2; maximum exponent of 5 is 2.
Use the smallest exponents to find the worked example GCD
A prime can appear in the GCD only if every number contains it. The exponent cannot exceed the smallest exponent among the numbers.
Minimum-exponent row
Only the two copies of 3 survive in every number.
Why the other primes disappear
At least one original number has exponent 0 for each of 2, 5, 7, 11, and 13. The minimum exponent in each of those columns is therefore 0.
Worked example 1 — GCD
Need a hint? Start here
A shared prime inventory cannot contain more copies than any one number supplies.
Check why it works
Only prime 3 has a positive minimum exponent, namely 2. Enter 3^2 (or 3² or 9) and numerical GCD 9. Any larger prime inventory would fail to divide at least one original number.
Use the largest exponents to find the worked example LCM
The LCM must contain enough copies of every prime to be divisible by each original number. That means taking the largest exponent in every column.
Maximum-exponent row
Every original factorization can be made by removing prime copies from this complete inventory. Reducing any maximum exponent would leave too few copies for at least one original number. This proves the common multiple is the least positive one.
Multiply carefully
Worked example 1 — LCM
Need a hint? Start here
Collect enough copies of each prime to cover the largest requirement.
Check why it works
The required powers are 2³,3³,5²,7,11,13. Exponent of 3: 3; exponent of 5: 2; LCM 5405400. Reducing any required power would lose divisibility by a original number.
Complete the worked example guided practice
Apply the same aligned-table method to 35, 98, and 112.
Guided Practice 1
| Number | 2 | 5 | 7 |
|---|---|---|---|
| 35 | 0 | 1 | 1 |
| 98 | 1 | 0 | 2 |
| 112 | 4 | 0 | 1 |
| Minimum exponents | 0 | 0 | 1 |
| Maximum exponents | 4 | 1 | 2 |
Enter both results
Need a hint? Start here
Insert zero for every missing prime before taking a minimum.
Check why it works
Minimum row 0,0,1; maximum row 4,1,2. GCD 7; LCM 2⁴×5×7²=3920.
Use a general prime-exponent laboratory
Enter two to six positive whole numbers. The page factors them, aligns every prime base, and builds the minimum and maximum exponent rows.
The laboratory accepts 2–6 integers from 1 through 1,000,000,000. The number 1 has no prime factors; its prime-exponent row contains zeros. If every input is 1, there are no prime columns, and both the GCD and LCM are 1. Write numbers without thousands separators, since commas separate different inputs.
Laboratory checkpoint
Answer for the fixed set 72, 90, and 120, even if you have explored another set in the lab.
Need a hint? Start here
Factor each input and align its prime columns. Check that the GCD divides each input.
Check why it works
Their minimum powers are 2×3; their maximum powers are 2³×3²×5. GCD 6; LCM 360.
Use exponent zero correctly when no prime is shared by all numbers
The chapter exercise gives a = 440, b = 126, and c = 825. Their prime columns overlap in pairs, but no prime appears in all three.
440
126
825
original Exercise 7 transfer
Need a hint? Start here
A prime shared by two numbers might still be absent from the third.
Check why it works
Every prime column has at least one zero. GCD 1; LCM 2³×3²×5²×7×11=138600.
Audit a new exponent table and catch common mistakes
A correct answer must use the minimum exponents for the GCD, maximum exponents for the LCM, and exponent zero for every missing prime.
Transfer factorizations
Error 1
“Use the largest exponents for the GCD.”
Repair: the GCD must divide every number, so use the smallest exponents.
Error 2
“Ignore a prime when one number does not contain it.”
Repair: write exponent 0. That zero may remove the prime from the GCD.
Error 3
“Once I have a product, no verification is needed.”
Repair: check that the GCD divides every number and every number divides the LCM.
Transfer checkpoint
Need a hint? Start here
Calculate the minimum and maximum separately for each prime column.
Check why it works
Minimum powers 2²×3; maximum powers 2⁴×3⁵×5²×7. GCD 12; LCM 680400.
Independent workshop
Correct all eight questions to complete the workshop.
1. gcd(18,30)
2. lcm(18,30)
3. gcd(35,98,112)
4. lcm(35,98,112)
5. If prime 13 is absent from a number’s factorization, its exponent in the aligned table is …
6. gcd(2520,14850,819)
7. lcm(2520,14850,819)
8. lcm(440,126,825)
Need a hint? Start here
Choose the method before calculating. Revisit the local hint for that method, then verify every condition in the question.
Check why it works
Include zeros wherever a prime is absent. Answers: 6; 90; 7; 3920; 0; 9; 5405400; 138600.
Objective exit ticket
Complete all ten missions, including 5 out of 5 on this ticket, to earn the certificate.
1. The GCD uses which exponents?
2. The LCM uses which exponents?
3. gcd(48,60,90)
4. lcm(48,60,90)
5. A missing prime has exponent …
Prime-Exponent GCD & LCM Architect
This certifies that the student can align prime factorizations, use zero exponents, and read both GCD and LCM from exponent columns.
Lesson 19.1 • Chapter 19
Optional reflection — not automatically graded
Need a hint? Start here
Choose the method before calculating. Revisit the local hint for that method, then verify every condition in the question.
Check why it works
Minimum powers 2×3; maximum powers 2⁴×3²×5. Smallest; largest; 6; 720; 0.