Why a Number Plus Its Reverse Is a Multiple of 11
Algebraic shortcut: ab + ba = 11(a+b)
47 + 74 = 121
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Split both: 47 is 40 + 7, and 74 is 70 + 4 — the same digits, swapped.
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Every digit lands once in the tens and once in the ones, so it is worth 11.
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So add the digits and multiply by 11: 4 + 7 = 11, and 11 × 11 = 121.
Why is a two-digit number plus its reverse always a multiple of 11?
Adding a two-digit number to its reverse always gives a multiple of 11. Take 47 + 74 = 121, which is 11 × 11. Split both numbers into tens and units: 47 is 40 + 7, and 74 is 70 + 4. Each digit appears once in the tens column and once in the units column, so the 4 is worth 40 + 4 = 44 and the 7 is worth 70 + 7 = 77. That makes the total 11 × 4 + 11 × 7, which is 11 × (4 + 7). So the sum is always 11 times the sum of the digits, and you can find it without adding the columns: 36 + 63 has digit sum 9, so it is 11 × 9 = 99.
When the digits add to 10 or more, the multiplication still works: 58 + 85 = 143, which is 11 × 13.
Your turn
Three to try in your head. Tap what you get.
78 + 87 = ?
34 + 43 = ?
45 + 54 = ?
All 36 shortcuts: the Magic Tricks library.