1
Two two-digit factors
Not answeredTwo two-digit positive integers have product 3927. Find the sum of the two integers.
sum
Lesson 18.7 — Chapter 18 Exercise
A complete English student edition of Test 18. Work through factor pairs, divisor counts, consecutive-number products, prime equations, trailing-zero completion, equal-product groups, and extended factorization searches.
Exercise directions
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Tips for this exercise
Factorization conventionsA divisor means a positive divisor. A prime factorization uses prime factors only. “Two-digit” means from 10 through 99, and a number ending in five zeros is divisible by 100,000.
List all possibilities in Question 8The age clues allow more than one age triple. This page asks for every possible unordered triple and every resulting sum.
Section I · Fill-in questions
Questions 1–10 · 6 points each · 60 points
2
Largest two-digit divisor
Not answeredThe number 151,200 has many positive divisors. What is its greatest two-digit divisor?
10–99
Optional factorization and search
3
Five consecutive odd numbers
Not answeredFive consecutive odd positive integers have product 328,185. What is the greatest of the five integers?
integer
Optional sequence and product check
4
A weighted equation in two primes
Not answeredThree times one prime plus two times another prime is 100. Find the product of the two primes.
product
Optional prime candidates
5
Three target shooters
Not answeredA, B, and C each take three shots at a target. Every shot score is a positive integer no greater than 10. For each person, the product of the three scores is 60. Their total scores satisfy A’s total > B’s total > C’s total; there are no ties. One of the nine shots scored exactly 4 points. Which person fired that shot?
Optional score triples and total comparison
6
Count the divisors of 126
Not answeredHow many positive divisors does 126 have?
divisors
Optional exponent calculation
7
Complete the product with five trailing zeros
Not answeredFind the least positive integer x such that
135 × 1925 × 486 × x
has its final five digits all equal to zero. In other words, the product must be divisible by 100,000.
positive integer
Optional counts of factors 2 and 5
8
Age product 90 — list every possibility
Not answeredThree children have positive whole-number ages in years, each younger than 10. Their ages have product 90. List every possible unordered age triple and every corresponding age sum. Ages may repeat; rearranging the same three ages does not make a new triple.
Why this wording is more explicitThe worked example asks for one sum, but the printed conditions do not determine a unique sum. Your response should include all possibilities supported by the clues.
Optional factor search
9
Split five numbers into equal-product groups
Not answeredPartition the five numbers below into two groups so that the product of the numbers in one group equals the product of the numbers in the other group.
5671415
Use every number exactly once, with at least one number in each group. The groups need not contain the same number of numbers. The order of the two groups does not matter.
Optional prime-exponent ledger
10
The unique prime from 200 through 220
Not answeredThere is exactly one prime number between 200 and 220, inclusive. What is it?
prime
Optional elimination table
Section II · Extended-response questions
Questions 11–13 · 20 points each · 60 points
11
Primes built from prime digit blocks
Not answeredA digit block is a group of neighboring digits kept in their original order. The worked example uses the prime 373 as an example. Keeping the digit order unchanged, its nonempty proper contiguous digit blocks are the one-digit blocks and the two-digit blocks:
3733773
Every one of these blocks is prime. Find all primes with two or more digits for which every nonempty contiguous digit block shorter than the whole numeral is also prime.
Self-contained interpretation of “split apart”A block must use consecutive digits in their original order. Repeated blocks may be written once in your final list of block values, but your proof must check every position.
Optional strategy hint
Start with the one-digit blocks, then list the allowed neighboring pairs. For a longer candidate, check every shorter block as well as the whole number.
Hint for proving your list is complete
Two neighboring three-digit blocks overlap in two positions. Use that overlap to decide whether a four-digit candidate could work. Any longer candidate must contain a four-digit block.
12
Three numbers centered on their average
Not answeredThree positive whole numbers have these properties:
The greatest is 6 more than the least.
The third number is the average of the greatest and the least.
The product of the three numbers is 42,560.
Find the three numbers.
13
Distribute the cards 1 through 9 by products
Not answeredNine cards are labeled with the numbers 1,2,3,4,5,6,7,8,9. A, B, and C each take exactly three cards.
The product of A’s three cards is 48.
The product of B’s three cards is 15.
The product of C’s three cards is 504.
Which three cards did each person take? Use every card exactly once.
Correction to the worked exampleThe lesson gives C’s product as 63, which is incompatible with using all nine cards exactly once. This exercise uses 504 so that the card-allocation problem is solvable. If you saved an earlier attempt, revisit Question 13 using this corrected target.