The Rule for 5: Half the Neighbor

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The rule for multiplying by 5 is simple: Take half of the neighbor, plus 5 if the digit itself is odd.

This rule is a variation of the rule for 10 (which is just adding a zero), but since 5 is half of 10, we deal with halves.

Example: Multiply 436 by 5

Add a leading zero: 0436.

  1. Units Digit (from 6): Neighbor is 0. Half of 0 is 0. Digit 6 is even, so no +5. Result is 0.
  2. Tens Digit (from 3): Neighbor is 6. Half of 6 is 3. Digit 3 is odd, so add 5. 3+5=83 + 5 = 8. Result is 8.
  3. Hundreds Digit (from 4): Neighbor is 3. Half of 3 is 1 (ignore remainder). Digit 4 is even, so no +5. Result is 1.
  4. Thousands Digit (from 0): Neighbor is 4. Half of 4 is 2. Digit 0 is even, so no +5. Result is 2.

The final answer is 2,180.

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Frequently Asked Questions

How does this rule work? It seems like magic.

It's based on the fact that multiplying by 5 is the same as multiplying by 10 and then dividing by 2 (e.g., N x 5 = N x 10 / 2). The Trachtenberg method cleverly turns this into a single pass. Taking 'half the neighbor' is part of the division by 2, and adding 5 for odd digits handles the remainders.

Is there really no carry in this rule?

Correct! This is what makes the rule for 5 so fast and simple. Each digit of the answer is calculated independently without affecting the next digit. This reduces the mental load significantly compared to traditional multiplication.

What about the very last digit of the number?

A common shortcut is to note that if the last digit of the original number is even, the answer will end in 0, and if it's odd, the answer will end in 5. Our general rule handles this automatically: the 'neighbor' of the last digit is 0, so its half is 0. You then add 5 only if the last digit is odd, giving you a final digit of 0 or 5.

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Frequently Asked Questions

How does this rule work? It seems like magic.

It's based on the fact that multiplying by 5 is the same as multiplying by 10 and then dividing by 2 (e.g., N x 5 = N x 10 / 2). The Trachtenberg method cleverly turns this into a single pass. Taking 'half the neighbor' is part of the division by 2, and adding 5 for odd digits handles the remainders.

Is there really no carry in this rule?

Correct! This is what makes the rule for 5 so fast and simple. Each digit of the answer is calculated independently without affecting the next digit. This reduces the mental load significantly compared to traditional multiplication.

What about the very last digit of the number?

A common shortcut is to note that if the last digit of the original number is even, the answer will end in 0, and if it's odd, the answer will end in 5. Our general rule handles this automatically: the 'neighbor' of the last digit is 0, so its half is 0. You then add 5 only if the last digit is odd, giving you a final digit of 0 or 5.