Arrange Prime Factors to Fit Digit Patterns
把质因数重新组合，满足数位与数字条件
In digit patterns, abc means the three-digit number 100a + 10b + c, not a × b × c. AB and CD likewise name two-digit numbers. Prime factorization creates a complete pool of possible factors. Digit length, distinct-letter rules, repeated blocks, leading-zero rules, and consecutive-prime conditions decide which arrangements actually fit.
Let factorization generate candidates; let digit conditions filter them
A correct product is only the first gate. A candidate must also satisfy every place-value and digit rule in the statement.
Digit length
A three-digit factor cannot be replaced by a two- or four-digit factor.
Different letters
Different letters represent different digits when the question explicitly requires this.
Leading zero
A first digit cannot be zero unless a fixed-width code is explicitly allowed.
Strategy check
Need a hint? Start here
Digit length puts bounds on each factor.
How can I check my reasoning?
Count positions a,b,c,d, then check lengths and the stated digit rule. Four positions; no leading zero; no, a correct product alone is insufficient.
Worked example 2 — solve abc × d = 1,995
Different letters represent different digits. The three-digit factor and the one-digit factor must use four different digits altogether.
Start with the prime inventory
The worked example regroups these prime factors into a three-digit number and a one-digit number.
Enter the unique valid expression
Need a hint? Start here
Try each possible one-digit divisor and check the quotient.
How can I check my reasoning?
d=1 gives a four-digit quotient; 3 and 5 give repeated digits. Only 285×7=1995 uses four different digits. Answers: 285; 7; 1995.
Use a complete three-digit × one-digit candidate laboratory
Testing only the first promising factor is not a proof. This laboratory checks every one-digit divisor from 1 through 9.
Three-digit × one-digit solver
Classify the worked example search
Need a hint? Start here
Check that all four digits differ after finding an exact product.
How can I check my reasoning?
Check all divisors 1 through 9, including unsuccessful cases. One valid arrangement: 285×7. The result is unique.
Guided Practice 2 — fit 1,995 into two two-digit factors
In AB × CD = 1,995, the four letters represent four different digits. Generate all factor pairs first; then apply the two-digit and distinct-digit filters.
Why one pair survives
Both factors must lie from 10 through 99. Then all four written digits must occur exactly once.
Order does not change the requested sum
21 × 95 and 95 × 21 use the same four digits. The worked example asks for their sum, so factor order is irrelevant.
Enter either factor order
Need a hint? Start here
Test both two-digit factors and inspect their digits.
How can I check my reasoning?
The two-digit pairs are 21×95 and 35×57; the second repeats 5. Enter 21 and 95 in either order. The digit sum is 2+1+9+5=17.
Translate a repeated digit block into multiplication
Treat the first three digits as one number and the final four digits as another. If the latter is ten times the former, the seven-digit number has a rigid form. In Missions 5–7, digits within the block may repeat: abc does not require a, b and c to differ.
Block form
Place-value identity
Writing the three-digit block twice gives 1000 × block + block. This creates 1001 copies of the block; appending zero multiplies by 10.
Repeated-block explorer
Complete the place-value check
Need a hint? Start here
abcabc0 is the block times (10,000 + 10).
How can I check my reasoning?
The multiplier is 10010=10×1001, and 1001=7×11×13. 10010; 13; 1231230. Repeated digits within abc are allowed.
Worked example 4 — recover the seven-digit phone number
The phone number is the product of eight consecutive primes, and its final four digits are ten times its first three digits.
Why the consecutive-prime window is forced
The number ends in 0, so its prime product contains both 2 and 5. A consecutive-prime list containing both must begin:
The five fixed prime factors already appear in the eight-prime product. The three remaining primes determine abc.
Recover the block
Verify the complete number
The final four digits are 9,690, exactly ten times 969.
Enter the worked example result
Need a hint? Start here
Divide out the fixed factors in 10,010, then match the remaining primes.
How can I check my reasoning?
The product includes 2, so its eight-prime window starts at 2 and ends at 19. After removing 2×5×7×11×13, the block is 3×17×19=969. Number: 9699690; largest prime: 19.
Guided Practice 4 — enumerate before claiming uniqueness
Find every seven-digit number meeting the conditions below. Use a complete consecutive-prime search to decide whether the answer is unique.
A seven-digit number has the form abcabc0 and equals four times the product of six consecutive primes.
State what the complete search proves
Need a hint? Start here
Check every consecutive-prime window until the product is too large.
How can I check my reasoning?
Stop once that result exceeds 9999999; later windows are larger. The two passing values are 1021020 and 6466460. Both have form abcabc0, so the answer is multiple.
Use a general factor-pattern studio
Digit length is itself a pattern. This transfer asks for two two-digit factors of 3,927, then lets you explore any product and digit-length combination.
Transfer problem
Two two-digit numbers have product 3,927. Find their sum.
Factor-pair pattern solver
With equal digit lengths, a pair and its reversal count once. With different lengths, the first and second factor must match their respective lengths. Repeated digits are allowed in the 3,927 transfer, including the two 7s in a factor.
Complete the transfer
Need a hint? Start here
Use factor pairs, then check digit lengths and the requested sum.
How can I check my reasoning?
3927=3×7×11×17. Generate factor pairs and keep two-digit factors. 51×77=3927. Smaller 51; larger 77; sum 128.
Digit-pattern and factor-arrangement workshop
Correct all eight responses 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?
Use the exact digit condition specified in each earlier problem. Answers: 4; 285; 7; 17; 10010; 969; 2; 128.
Objective exit ticket
Complete all ten missions, including 5 out of 5 on this exit ticket, to earn the certificate.
Prime-Factor Digit-Pattern Architect
This certifies that
can generate factor candidates, apply digit-length and distinct-digit constraints, translate repeated blocks into multiplication, search consecutive-prime windows, and identify non-unique original conditions.
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?
For 1728, the valid distinct-digit products are 864×2, 576×3 and 432×4. Answers: 13; 2342340; 21; 143; 3.