Unit 3 · Topic 01 · Playing with Numbers
Ms. Rao offered a prize: whoever could tell her, without dividing, whether 4,928 was divisible by 8, in under five seconds, first.
Kabir grabbed a pencil to do long division. Anaya just looked at the last three digits.
Anaya called out "yes!" before Kabir had written the first line.
"4,928 — divisible by 8, yes or no, five seconds, go!" Ms. Rao said. Kabir immediately started dividing 4928 by 8 on paper. Anaya instead looked only at the last three digits, 928, and checked in her head whether THAT smaller number divided by 8. It did: 928 / 8 = 116. "Yes!" she called, well before Kabir's long division was even half finished.
"How did you do that so fast?" Kabir asked, stunned. Ms. Rao explained: "For divisibility by 8, you never need to check the whole number — just its last three digits. If those divide evenly by 8, so does the whole thing, because every thousand is automatically divisible by 8." She showed a similar shortcut for 11: subtract the sum of digits in odd positions from the sum of digits in even positions; if the result is 0 or a multiple of 11, the whole number is divisible by 11.
For 7, she admitted there wasn't as clean a shortcut, but showed a working method: double the last digit, subtract it from the rest of the number, and check if THAT smaller result divides by 7. Kabir tried it on 483: last digit 3 doubled is 6, and 48-6=42, which is 7x6. So 483 is divisible by 7 — confirmed instantly by long division too.
"None of these shortcuts skip real math," said Ms. Rao. "They just replace one big division with a smaller, faster check that gives the exact same answer." Kabir tried the 8-rule on 5,000 + 128 = 5128: last three digits 128, and 128/8=16 exactly, so 5128 divides by 8 too — no long division needed, and no doubt about the answer either.
Divisibility by 8: check only the last three digits of the number. If those three digits form a number divisible by 8, so is the whole number, since every multiple of 1000 is automatically divisible by 8.
Divisibility by 11: alternately add and subtract the digits from right to left (or sum digits in odd positions and even positions separately and subtract). If the result is 0 or a multiple of 11, the number is divisible by 11.
Divisibility by 7: double the last digit, subtract it from the remaining number, and repeat if needed. If the final result is 0 or a multiple of 7, the original number is divisible by 7.
Pick any 4-digit number from a receipt, phone number, or house number. Test it for divisibility by 8 using only its last three digits, then check with a calculator.
Apply the 11-rule to your own birth year or any 4-digit number you like, and confirm your answer with division.
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