Problem
Number satisfying sum and product clues
The month and the day add to 23 and multiply to 120. The month is a whole number from 1 to 12. The day must exist in that month. Find the birthday.
Reasoning · Grade 3-1 Multiplication
Your answer
Watch out Finding 8 and 15 feels done, but 15 cannot be a month, so it reads August 15.
How to solve
Strategy Make a Systematic List — Two numbers with a fixed product 120 come from a short list of factor pairs. Listing the factor pairs of 120 and checking which pair adds to 23 finds the answer directly, with no algebra.
1STEP 1
List factor pairs of 120
Pairs multiplying to 120 run from 1 and 120 to 10 and 12 — eight of them.
120 = 1×120 = 2×60 = 3×40 = 4×30 = 5×24 = 6×20 = 8×15 = 10×12
Finding factor pairs is just listing which times-table facts make 120 - a small, finite list.
3.OA.B.6Make A Systematic ListListing the factor pairs of 120 turns an open search into a short, finite list.
Why?
Every number that divides 120 hands over its partner at the same time, so working upward from 1 catches every pair and misses none.
🧱Divisors come in pairsEach divisor hands over its partner for free, so divisors pair off two by two unless one is its own partner.
Why?
120 breaks into primes in only one way, so the divisors it can have are fixed and the list cannot run on forever.
🧱Every number has one prime factorizationKeep splitting a number into factors and you always land on the same bag of primes, no matter which split you tried first.
2STEP 2
Find the pair that adds to 23
Only 8 and 15 also add to 23.
8 + 15 = 23
Adding each pair is quick mental arithmetic, and only one pair lands on 23.
3.NBT.A.2Guess And Check3STEP 3
Assign month and day
15 cannot be a month, so it is August 15.
month = 8, day = 15
Only one of the two numbers can be a month (1-12), which removes the ambiguity.
3.OA.A.4Make A Systematic ListAnswer
August 15
Check both conditions: 8 + 15 = 23 and 8 x 15 = 120. The month 8 is within 1-12 and August 15 is a real date, so the answer fits every requirement.
Takeaway
List the times-table pairs that make 120, then check which pair adds to 23 - pure Grade 3 multiplication facts!
- List factor pairs of 120
- Find the pair that adds to 23
- Assign month and day