Title: Gmat Quant Number Properties Tricks — LearnCoachAssist

Q: Divisibility by 3?
A: Sum of digits divisible by 3. E.g., 4521 → 4+5+2+1=12 → divisible.

Q: Divisibility by 4?
A: Last two digits divisible by 4. E.g., 1532 → 32 ÷ 4 = 8.

Q: Divisibility by 6?
A: Even AND divisible by 3.

Q: Divisibility by 8?
A: Last three digits divisible by 8.

Q: Divisibility by 9?
A: Sum of digits divisible by 9.

Q: Divisibility by 11?
A: Alternating sum of digits divisible by 11 (positive at odd positions, negative at even).

Q: Difference between integers and whole numbers?
A: Whole numbers: 0, 1, 2, 3…
Integers: …, -2, -1, 0, 1, 2…

Q: Prime numbers — 1 included?
A: No. Smallest prime is 2; it's also the only even prime.

Q: How many prime numbers under 20?
A: 8: 2, 3, 5, 7, 11, 13, 17, 19.

Q: Number of divisors trick?
A: Prime-factorize, add 1 to each exponent, multiply. 12 = 2² × 3¹ → (2+1)(1+1) = 6 divisors.

Q: Sum of digits of 5^n cycle?
A: 5^1=5, 5^2=25, 5^3=125, 5^4=625, 5^5=3125. Last digit always 5.

Q: Last digit of 7^n?
A: Cycle of 4: 7, 9, 3, 1. So 7^25 → 25 mod 4 = 1 → last digit 7.

Q: Product of consecutive integers includes…?
A: Always includes a multiple of any small prime up to the number of consecutive terms. E.g., k(k+1)(k+2) always divisible by 6.

Q: Sum of first n positive integers?
A: n(n+1)/2.

Q: Sum of squares 1²+2²+...+n²?
A: n(n+1)(2n+1)/6.

Q: Sum of cubes 1³+2³+...+n³?
A: (n(n+1)/2)² — square of the sum.

Q: Sum of first n odd numbers?
A: n².

Q: Even × any = ?
A: Even.

Q: Odd × odd = ?
A: Odd.

Q: Odd + odd = ?
A: Even.

Q: If x and y are integers and 2x + 3y is even, what's true?
A: y must be even (3y has same parity as y; for sum to be even, 3y must be even).

Q: GCD via prime factorization?
A: Take each common prime to the lowest exponent. 12=2²×3, 18=2×3² → GCD=2×3=6.

Q: LCM via prime factorization?
A: Each prime to the highest exponent. 12, 18 → LCM = 2²×3² = 36.

Q: LCM × GCD = ?
A: a × b. So LCM(a,b) = ab / GCD(a,b).

Q: Remainder properties?
A: Distributive over multiplication: (a×b) mod n = ((a mod n)(b mod n)) mod n.

Q: If x leaves remainder 3 when divided by 7, x² leaves what?
A: 3² = 9 mod 7 = 2.

Q: Squares mod 4?
A: Always 0 or 1.

Q: Number is divisible by both 4 and 6?
A: Need LCM(4,6)=12. Not 24 — common mistake.

Q: Sum of digits modular trick?
A: Number ≡ sum of digits (mod 9) — basis of divisibility tests.

Q: Decimal to fraction: 0.3333…?
A: 1/3.

Q: Recognize 0.1666…?
A: 1/6.

Q: Convert percent to decimal quickly?
A: Move decimal two places left. 12.5% → 0.125.

Q: Successive percent change pitfall?
A: 20% increase then 20% decrease ≠ original. 100 → 120 → 96. Always multiply: 1.2 × 0.8 = 0.96.

Q: Average speed formula for two equal distances?
A: 2ab/(a+b) (harmonic mean), where a and b are the two speeds.

Q: If positive, x > 1 vs 0 < x < 1 — x²?
A: x > 1: x² > x.
0 < x < 1: x² < x. Easy trap.

Q: Negative number squared?
A: Positive: (-3)² = 9. But -3² = -9 (no parentheses → unary minus).

Q: Number of factors of 360?
A: 360 = 2³ × 3² × 5 → (3+1)(2+1)(1+1) = 24.

Q: Trailing zeros in n!?
A: Count factors of 5 (since 2s are plentiful): floor(n/5) + floor(n/25) + floor(n/125) + ...

Q: Trailing zeros in 25!?
A: 25/5 + 25/25 = 5 + 1 = 6.

Q: Average of consecutive integers from a to b?
A: (a+b)/2 — also the median.

Q: Number of integers from a to b inclusive?
A: b - a + 1.

Q: Divisibility by 2?
A: Last digit is even (0, 2, 4, 6, 8).

Q: Divisibility by 5?
A: Last digit is 0 or 5.

Q: Divisibility by 7 (quick test)?
A: Double the last digit, subtract from the rest. If the result is divisible by 7 (or 0), so is the original. E.g., 203 → 20 − 6 = 14 ✓.

Q: Divisibility by 12?
A: Divisible by both 3 and 4.

Q: Divisibility by 15?
A: Ends in 0 or 5 AND digit sum divisible by 3.

Q: Divisibility by 25?
A: Last two digits are 00, 25, 50, or 75.

Q: Divisibility by 125?
A: Last three digits are 000, 125, 250, 375, 500, 625, 750, or 875.

Q: Quick check: is 143 divisible by 11?
A: Alternating sum: 1 − 4 + 3 = 0 → yes. (143 = 11 × 13.)

Q: Is 105 divisible by 7?
A: 10 − (2×5) = 0 → yes. (105 = 7 × 15.)

Q: Is 1458 divisible by 9?
A: Digit sum 1+4+5+8 = 18 → yes. Also divisible by 3 and by 2.

Q: Smallest positive integer divisible by every integer from 1 to 10?
A: 2520 = LCM(1,…,10).

Q: LCM(12, 15, 20)?
A: Prime factors: 12=2²·3, 15=3·5, 20=2²·5 → 2²·3·5 = 60.

Q: GCD(48, 180)?
A: 48=2⁴·3, 180=2²·3²·5 → 2²·3 = 12.

Q: Are 18 and 35 coprime?
A: Yes — share no prime factors, so GCD = 1 and LCM = 18·35 = 630.

Q: Euler's totient φ(36)?
A: 36=2²·3² → φ=36·(1−½)·(1−⅓) = 36·½·⅔ = 12.

Q: Sum of positive divisors of 28?
A: 1+2+4+7+14+28 = 56 (a perfect number: equals 2·28).

Q: Composite vs. prime?
A: Composite has a divisor other than 1 and itself; prime has none. 1 is neither.

Q: Twin primes under 30?
A: (3,5), (5,7), (11,13), (17,19), (29,31) — five pairs.

Q: Only even prime?
A: 2.

Q: Consecutive integers that are both prime?
A: Only (2, 3).

Q: Smallest composite number?
A: 4.

Q: How many primes between 50 and 70?
A: Four: 53, 59, 61, 67. (51=3·17, 55=5·11, 57=3·19, 63=7·9, 65=5·13, 69=3·23.)

Q: Is 91 prime?
A: No. 91 = 7 × 13.

Q: Largest prime under 100?
A: 97.

Q: Fundamental Theorem of Arithmetic?
A: Every integer > 1 has a unique prime factorization (up to order).

Q: Largest power of 2 dividing 100!?
A: ⌊100/2⌋+⌊100/4⌋+⌊100/8⌋+⌊100/16⌋+⌊100/32⌋+⌊100/64⌋ = 50+25+12+6+3+1 = 97. So 2⁹⁷.

Q: Highest power of 3 dividing 50!?
A: ⌊50/3⌋+⌊50/9⌋+⌊50/27⌋ = 16+5+1 = 22. So 3²².

Q: What is 0! ?
A: 1 (by definition).

Q: Last two digits of 37⁴⁰?
A: 37²≡69, 37⁴≡61 (mod 100); 37⁴⁰=(37⁴)¹⁰≡61¹⁰. Since 61⁵≡1 (mod 100), 61¹⁰≡1. So last two digits: 01.

Q: Last digit of 3¹⁰⁰?
A: Cycle 3,9,7,1 of length 4. 100 mod 4 = 0 → last digit 1.

Q: Last digit of 2⁵⁰?
A: Cycle 2,4,8,6 of length 4. 50 mod 4 = 2 → last digit 4.

Q: Last digit of 13⁷⁹?
A: Same cycle as 3: 3,9,7,1. 79 mod 4 = 3 → last digit 7.

Q: Sum of first 20 even numbers?
A: 2(1+2+…+20) = 2·210 = 420.

Q: Sum of first 100 multiples of 7?
A: 7·(1+2+…+100) = 7·5050 = 35350.

Q: Sum 1+2+…+100?
A: 5050.

Q: Sum 11+12+…+20?
A: Arithmetic series with 10 terms, average 15.5 → 155.

Q: Sum of multiples of 3 from 3 to 99?
A: 33 terms, average 51 → 1683.

Q: Sum of first 50 odd numbers?
A: 50² = 2500.

Q: Sum of first 30 even numbers?
A: 2·465 = 930. Also = 30·31 = 930.

Q: Arithmetic mean of 12, 18, 24, 30?
A: (12+18+24+30)/4 = 21.

Q: Geometric mean of 4 and 36?
A: √144 = 12.

Q: Harmonic mean of 2 and 8?
A: 2/(1/2 + 1/8) = 16/5 = 3.2.

Q: Median of 3, 8, 11, 17, 25?
A: 11 (middle value when sorted).

Q: Mode of 2, 5, 5, 8, 9, 9, 9, 10?
A: 9 (appears 3 times).

Q: Range of −3, 7, 12, 0, 5?
A: 12 − (−3) = 15.

Q: What is a perfect square?
A: An integer of the form n² (n ≥ 0): 0, 1, 4, 9, 16, 25, …

Q: What is a perfect cube?
A: An integer of the form n³: 0, 1, 8, 27, 64, 125, 216, …

Q: Perfect squares between 50 and 200?
A: 64, 81, 100, 121, 144, 169, 196 — seven.

Q: Perfect squares are always... ?
A: Congruent to 0 or 1 mod 4.

Q: Parity: even + even?
A: Even.

Q: Parity: odd + even?
A: Odd.

Q: Parity: odd × even?
A: Even.

Q: Parity of 17 × 23 + 9?
A: Odd × odd = odd; odd + odd = even.

Q: Parity of 14n + 1 for any integer n?
A: Odd (even + 1).

Q: Parity of n² + n for any integer n?
A: Even (one of n, n+1 is even; their product is even).

Q: Parity of n³ − n?
A: Always even (n(n−1)(n+1) is divisible by 2 and 3).

Q: n(n+1) is always... ?
A: Even (product of two consecutive integers).

Q: n(n+1)(n+2) is always divisible by... ?
A: 6. Among three consecutive integers, one is a multiple of 3 and at least one is even.

Q: n(n+1)(n+2)(n+3) is always divisible by... ?
A: 24 (= 4!).

Q: What is a factorial?
A: n! = n·(n−1)·…·2·1. 0! = 1.

Q: 10! = ?
A: 3628800.

Q: C(10,3) (binomial coefficient)?
A: 10!/(3!·7!) = 120.

Q: Sum of digits of 7⁵⁰ (mod 9)?
A: 7³≡1 (mod 9); 50 mod 3 = 2 → 7⁵⁰ ≡ 7² = 49 ≡ 4 (mod 9).

Q: Remainder when 2¹⁰⁰ is divided by 9?
A: 2³≡8≡−1; 2⁹⁹≡(−1)³³=−1; 2¹⁰⁰≡−2≡7. Remainder 7.

Q: Remainder when 5²³ is divided by 13?
A: 5²=25≡−1; 5²²≡1; 5²³≡5. Remainder 5.

Q: Remainder when 7⁶⁰⁰ is divided by 5?
A: 7≡2; 2⁴≡1; 600 mod 4 = 0 → 7⁶⁰⁰≡1.

Q: Remainder when 3⁵⁰ is divided by 7?
A: 3⁶≡1; 50 mod 6 = 2 → 3⁵⁰≡3²=9≡2.

Q: What is a unit / ones digit?
A: The rightmost digit of an integer, equal to value mod 10.

Q: Pattern of last two digits of n²?
A: Squares end in 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96 (mod 100).

Q: Why no squares end in 2, 3, 7, or 8?
A: The last digit of a square is always 0, 1, 4, 5, 6, or 9.

Q: Is 1234567 a perfect square?
A: No — mod 4: 1234567 mod 4 = 3, but squares mod 4 are 0 or 1.

Q: Difference of squares: x² − y² factors as... ?
A: (x−y)(x+y).

Q: Sum of squares: x² + y² factors over... ?
A: Doesn't factor over integers; over complex: (x+iy)(x−iy).

Q: Can a² + b² be a perfect square?
A: Yes — Pythagorean triples. E.g., 3² + 4² = 5².

Q: Smallest Pythagorean triple with integer sides?
A: (3, 4, 5).

Q: General form of primitive Pythagorean triple?
A: a = m²−n², b = 2mn, c = m²+n² (coprime m>n, opposite parity).

Q: Why is √2 irrational?
A: Suppose √2 = p/q in lowest terms → 2q² = p² → p even → p = 2k → q² = 2k² → q even — contradiction.

Q: How many digits does 50! have?
A: ⌊log₁₀(50!)⌋ + 1 = 65.

Q: What is a Mersenne prime?
A: A prime of form 2ᵖ−1 with p prime. E.g., 3, 7, 31, 127, 8191.

Q: What is a Fermat prime?
A: A prime of form 2^(2ⁿ)+1. Known: 3, 5, 17, 257, 65537.

Q: What is a Sophie Germain prime?
A: A prime p such that 2p+1 is also prime. E.g., 2, 3, 5, 11, 23, 29, …

Q: What does it mean for a number to be abundant?
A: Sum of proper divisors exceeds the number. E.g., 12: 1+2+3+4+6 = 16 > 12.

Q: What does it mean for a number to be deficient?
A: Sum of proper divisors is less than the number. E.g., 8: 1+2+4 = 7 < 8.

Q: What does it mean for a number to be perfect?
A: Sum of proper divisors equals the number. E.g., 6, 28, 496, 8128.

Q: What is an amicable pair?
A: Two numbers where each is the sum of proper divisors of the other. Smallest: (220, 284).

Q: What is a Harshad (Niven) number?
A: An integer divisible by the sum of its digits. E.g., 18: 1+8=9, 18/9=2 ✓.

Q: What is a palindrome number?
A: Reads the same forwards and backwards. E.g., 121, 1331, 12321.

Q: What is a Kaprekar number?
A: A number whose square splits into two parts summing to the original. E.g., 45²=2025 → 20+25=45.

Q: What is an Armstrong (narcissistic) number?
A: Equal to the sum of its digits each raised to the power of the number of digits. E.g., 153 = 1³+5³+3³.

Q: Automorphic number?
A: Ends in its own square. 5²=25, 6²=36, 25²=625, 76²=5776.

Q: What is a repunit?
A: A number consisting only of the digit 1: 1, 11, 111, 1111, …

Q: Why is 111111 = 3·7·11·13·37?
A: 111111 = 111·1001 = (3·37)·(7·11·13).

Q: Express 0.123̄ as a fraction?
A: Let x = 0.123̄. 1000x − x = 123 → x = 123/999 = 41/333.

Q: Express 0.1̄ as a fraction?
A: 1/9.

Q: Express 0.142857̄ as a fraction?
A: 142857/999999 = 1/7.

Q: Express 0.625 as a fraction?
A: 625/1000 = 5/8.

Q: What is 7% of 250?
A: 17.5. (1% of 250 = 2.5, times 7.)

Q: What is 15% of 80?
A: 12. (10% of 80 = 8, plus half = 4.)

Q: 15 is what percent of 75?
A: 20%.

Q: 25 is what percent of 200?
A: 12.5%.

Q: A price rises 10% then falls 10%. Net change?
A: 1.1·0.9 = 0.99 → 1% decrease.

Q: A price rises 50% then falls 50%. Net change?
A: 1.5·0.5 = 0.75 → 25% decrease.

Q: Population grows 3% per year. Factor after 10 years?
A: 1.03¹⁰ ≈ 1.3439.

Q: Rule of 72: doubling time at r% per year ≈ ?
A: 72/r years. E.g., 6% growth doubles in ~12 years.

Q: Discount chain: 20% off, then 10% off sale price. Net discount?
A: 1 − 0.8·0.9 = 0.28 → 28% off.

Q: Markdown then tax: item discounted 20%, then 8% sales tax. Original $100 → final?
A: 100·0.8·1.08 = $86.40.

Q: Mixture: 6 kg of $4/kg coffee blended with 4 kg of $8/kg coffee. Average price/kg?
A: (6·4 + 4·8)/10 = $5.60/kg.

Q: Alligation: 10 L of 30% acid, 40 L of 60% acid. Final concentration?
A: (300+2400)/50 = 54%.

Q: Mixing 5 L milk with 5 L water, then removing 5 L of mixture. Milk left?
A: 5·(5/10) = 2.5 L.

Q: Speed × time = distance. Time to cover 180 km at 60 km/h?
A: 3 hours.

Q: Two trains 240 km apart approach at 40 km/h and 80 km/h. Time to meet?
A: 240/120 = 2 hours.

Q: Two trains start at the same point, opposite directions at 50 and 70 km/h. Distance after 3 h?
A: 360 km.

Q: Same direction, 60 km/h and 40 km/h, starting 100 km apart. Time for faster to catch slower?
A: 100/20 = 5 hours.

Q: Boat in still water 10 km/h, river flows 4 km/h. Effective speed downstream?
A: 14 km/h.

Q: Effective speed upstream (boat 10, current 4)?
A: 6 km/h.

Q: Round trip: downstream d at v+u, upstream at v−u. Average speed?
A: (v²−u²)/v.

Q: Two taps fill a tank in 6 min and 10 min; both open?
A: 1/6+1/10 = 4/15 per min → 15/4 = 3 min 45 s.

Q: Work problem: A does job in 12 days, B in 18 days. Together?
A: 1/12+1/18 = 5/36 per day → 36/5 = 7.2 days.

Q: A and B do a job in 8 days; A alone in 12. B alone?
A: 1/B = 1/8−1/12 = 1/24 → 24 days.

Q: Pipes: one fills in 6 min, another empties in 9 min. Net time to fill?
A: 1/6−1/9 = 1/18 per min → 18 min.