Concept 8 of 8Foundation

📌 Formula Sheet — Number Play

Rules and shortcuts — with the reason each one works.

Place value = digit × 10^(position from ones)

Means: ones place = 10⁰, tens = 10¹, hundreds = 10², …Use when: reading large numbers, expanded form

Why: Our system is base 10. Each shift left multiplies the weight by 10. In 472, the 4 is worth 4 × 100 because it's 2 positions left of ones.

⚠ Trap: Face value (just the digit) ≠ place value (digit × weight).

Divisibility rules

by 2: last digit evenby 5: last digit 0 or 5by 10: last digit 0by 3: digit sum ÷ 3by 9: digit sum ÷ 9by 4: last 2 digits ÷ 4by 8: last 3 digits ÷ 8by 6: ÷2 AND ÷3

Why (by 3 and 9): 10 leaves remainder 1 when divided by 3 (and by 9). So every digit contributes itself to the remainder. Therefore whole number mod 3 = (digit sum) mod 3.

⚠ Trap: Divisibility by 6 is NOT the same as "by 2 OR by 3" — it needs BOTH.

Rounding: look at the digit JUST RIGHT of the rounding place

Rule: ≥ 5 round up, < 5 round downUse when: estimating, simplifying

Why: 5 is halfway — by convention we round up so estimates don't systematically under-count.

⚠ Trap: When rounding 4.49 to nearest whole, answer is 4 (NOT 5) — look at ONE digit right, not all remaining.

Digital root: keep adding digits until single digit

Means: e.g., digital root of 2934 = 2+9+3+4 = 18 → 1+8 = 9Use when: checking divisibility by 9

Why: Because 10 ≡ 1 (mod 9), the digit sum gives the same remainder as the number. Repeated digit summing reduces to that remainder.

⚠ Trap: Digital root 0 and 9 both mean divisible by 9 (convention: root of 0 is 0).

Roman numerals: I=1, V=5, X=10, L=50, C=100, D=500, M=1000

Rule: add left-to-right; smaller BEFORE larger means subtractUse when: reading old chapter numbers, clock faces, dates

Why subtraction? Romans avoided four symbols in a row. "IIII" became "IV" to save ink — 1 before 5 means 5−1.

⚠ Trap: Only subtract ONE smaller symbol before a larger one. "IIX" for 8 is wrong — write "VIII".

💡 Tip:Digit-sum divisibility (by 3 and 9) is the most common shortcut in olympiads.

▶Prefer a video? Open YouTube search for “number system rules class 6 revision”↗