Mental Math · Divisibility
Big numbers look scary to divide, but every divisor only peeks at one tiny tail — try each problem first, then learn the test that cracks it in your head.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: Every test isolates the smallest piece the divisor can see — the rest is a power of 10 it already divides — and the same computation that says ‘divisible’ hands you the remainder for free.
★ · The Divisor Only Sees the Tail
Is 3,752 divisible by 5? By 4? By 8? By 3? By 9? And does 7,920 pass every test for 2, 3, 4, 5, 6, 8, 9, and 10?
L0 · Run the Rules — Each Divisor Reads Its Own Tail
Which of 2, 3, 4, 5, 6, 8, 9, 10 divide 7,740? And which of them divide 5,832?
L1 · Turn the Rule into an Equation
Find the digit A so that 46A1 is divisible by 9. Then find digits A and B so that 257AB is divisible by 36, making A + B as large as possible.
L2 · The Alternating Swing (÷11)
Find the digit A so that 8A35 is divisible by 11. And what is the remainder when 918,273 is divided by 11?
L3 · One Thousand Is One Short
Is 12,345,678 divisible by 7? And what is the remainder when it is divided by 13?
L4 · Coprime Pieces
Is 18 divisible by 12? Then find every (A, B) so that 7A2B is divisible by 72.
L5 · Everything, Combined
Find digits A and B so that 5A41B2 is divisible by 88, and give the largest such number. Then: is 638,638 divisible by 7, 11, and 13, and why does every number of the form abcabc behave the same way?
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
★ · The Divisor Only Sees the Tail
Peek the trick — The Divisor Only Sees the Tail
If a divisor splits a power of 10 (2, 5, 10 split 10; 4 and 25 split 100; 8 and 125 split 1,000), it can only read the matching tail of the number — the front is built from that power of 10, so it is along for the ride. For 3 and 9, every power of 10 leaves remainder 1, so the divisor reads the digit sum. And N mod d, the remainder, falls out of that same reading for free.
L0 · Run the Rules
Peek the trick — Run the Rules
If a number faces a whole list of divisors at once, run each one's rule on the piece it can see — last digit for 2, 5, 10, last two for 4, last three for 8, the digit sum for 3 and 9 — and let 6 fall out of (÷2 AND ÷3). Watch the traps: passing 3 does not mean passing 9, and the 4-test never looks past the last two digits.
L1 · Turn the Rule into an Equation
Peek the trick — Turn the Rule into an Equation
If a digit is missing, take the piece the divisor reads — the digit sum for 3 and 9, the last two digits for 4 — set it equal to a multiple of the divisor, and solve for the blank. For a composite divisor, split it into coprime factors, write BOTH conditions, intersect, then optimize last.
L2 · The Alternating Swing
Peek the trick — The Alternating Swing
If you tag the digits from the right with alternating signs (+, −, +, −, …) and add them, then 0 or any multiple of 11 means the number is divisible by 11 — because 10 ≡ −1 (mod 11), so each step left flips the sign. Whatever that alternating sum leaves mod 11 IS the remainder.
L3 · One Thousand Is One Short
Peek the trick — One Thousand Is One Short
If you split a number into 3-digit blocks from the right and alternately subtract and add them (… +ghi −def +abc), then that one small block-sum has the same remainder mod 7 — and mod 13 — as the whole number. Why: 1001 = 7 × 11 × 13, and 1000 is just one short of 1001, so every thousand you climb flips the sign.
L4 · Build It from Coprime Pieces
Peek the trick — Build It from Coprime Pieces
If the divisor is composite, split it into factors that share no common factor (coprime), apply each factor's rule, and the number must pass them ALL. A split that shares a factor (12 ≠ 2 × 6) leaks false positives. The tail-siblings extend the same reflex: 25 reads the last two digits, 125 reads the last three, and 101 tags 2-digit blocks +, −, + from the right.
L5 · Everything, Combined
Peek the trick — Everything, Combined
No new rule — the capstone fuses every move at once. If a divisor factors into coprime pieces, split it, isolate the deciding digits for EACH piece, and satisfy all the conditions together. And read structure before you ever divide: abab = ab × 101, and abcabc = abc × 1001 = abc × 7 × 11 × 13.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
