A short test on the digits tells you whether a number divides evenly, without doing the division. A number is divisible by 3 when its digits add to a multiple of 3, by 4 when its last two digits are, by 5 when it ends in 0 or 5, and by 9 when its digit sum is.
You'll recognise it when…
“which of these numbers is divisible by”
“a number leaves no remainder when divided”
“find the missing digit that makes it a multiple of”
●●○4 points, difficulty 2 of 34 ptLevel 3–4about 60s
The four-digit number 27_4 is divisible by 3. Which digit could go in the gap?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Add the digits you already know.
2The strategy
A number is divisible by 3 exactly when its digits add to a multiple of 3. Add the three known digits, then ask what the missing digit must be to push the total onto a multiple of 3.
3The full solution
2 + 7 + 4 = 13. The next multiple of 3 above 13 is 15, so the missing digit is 2. (8 would also work, but it is not offered.)
Solution— no shortcut on this one
The test for 3 is the digit sum. The known digits give 2 + 7 + 4 = 13, so the full digit sum is 13 plus the missing digit. For that to be a multiple of 3 the missing digit must be 2, giving 15, or 5, giving 18, or 8, giving 21. Only 2 is among the choices. The transferable idea: convert a divisibility question into a question about the digit sum.
Why this is on the test: Divisibility rules are pure speed on a timed paper, and the common failure is testing the missing digit instead of the whole digit sum.
Where it goes wrong
Each of these is written the way a student would say it to themselves.
“I tested divisibility by 6 by checking only that the number was even.”
“I added the digits to test for 3 but then checked the sum against 3 instead of asking if the sum is a multiple of 3.”
“I used the last digit to test for 4, but 4 needs the last two digits.”
Every problem in this skill
●○○3 points, difficulty 1 of 33 ptOne step from a skill you know.
Nothing at this tier yet for NUM-04.
●●○4 points, difficulty 2 of 34 ptTwo steps, or one step plus a trap.