Depth · one concept from first principles · step 2 of 4
Same shortfall turns remainders into one LCM
A number leaves 2 when divided by 3 and 4 when divided by 5. Find what it leaves when divided by 15.
Number systemExpressions
Your answer
How to solve
Strategy Change Focus / Count the Complement — Instead of looking at what is left over, look at what is missing. Both remainders are exactly 1 short of the divisor, so adding 1 makes the number a common multiple of 3 and 5.
1STEP 1
Look at the shortfall
A remainder of 2 out of 3 is 1 short of the next multiple of 3; a remainder of 4 out of 5 is 1 short of the next multiple of 5.
x = 3a + 2 → x + 1 = 3(a + 1), x = 5b + 4 → x + 1 = 5(b + 1)
Adding 1 makes x a multiple of both 3 and 5.
7.EE.A.2Change Focus Count The Complement2STEP 2
Use the least common multiple
A common multiple of 3 and 5 is a multiple of their least common multiple, 15.
x + 1 = 15k
x is one less than a multiple of 15.
6.NS.B.4Convert To AlgebraOne more than the number is a multiple of .
Why?
The number is some full groups of plus left over, and more fills that leftover into one more full group of ; the same turns the leftover into one more full group of .
🧱Division with remainderAny number splits into a whole count of equal groups plus a leftover, and the leftover is always smaller than the group size.
Why?
So the number plus is on the list of multiples of and on the list of multiples of , and the numbers on both lists are exactly the multiples of , where the two lists first agree.
🧱Common multiples are the multiples of the least oneThe numbers that appear on two multiple lists at once are exactly the multiples of the first number where the lists agree, so nothing between them can be shared.
3STEP 3
Read off the remainder
Rewrite x so that the part left over is between 0 and 14.
x = 15k - 1 = 15(k - 1) + 14
Divided by 15, the remainder is 14.
7.EE.A.2Convert To Algebra4STEP 4
List a few to confirm
The numbers are 14, 29, 44, …: each is 14 more than a multiple of 15.
14 = 15 × 0 + 14, 29 = 15 × 1 + 14
Every such number leaves 14.
6.NS.B.4Make A Systematic ListAnswer
14
Take 29: 29 = 3 × 9 + 2 and 29 = 5 × 5 + 4, as required, and 29 = 15 × 1 + 14.
Takeaway
When the remainders differ, check the shortfall -- if it is the same, add it back and use the LCM.
- Look at the shortfall
- Use the least common multiple
- Read off the remainder
- List a few to confirm
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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