Gmat Quant Number Properties Tricks Practice Exam

Question 1 Gmat Quant Number Properties Tricks

n(n+1)(n+2)(n+3) is always divisible by... ?

24 (= 4!). 7·(1+2+…+100) = 7·5050 = 35350. 4. Last two digits are 00, 25, 50, or 75.

Question 2 Gmat Quant Number Properties Tricks

Euler's totient φ(36)?

1.1·0.9 = 0.99 → 1% decrease. 12.5%. 36=2²·3² → φ=36·(1−½)·(1−⅓) = 36·½·⅔ = 12. 7≡2; 2⁴≡1; 600 mod 4 = 0 → 7⁶⁰⁰≡1.

Question 3 Gmat Quant Number Properties Tricks

What is an Armstrong (narcissistic) number?

Even AND divisible by 3. Last three digits divisible by 8. Equal to the sum of its digits each raised to the power of the number of digits. E.g., 153 = 1³+5³+3³. A prime p such that 2p+1 is also prime. E.g., 2, 3, 5, 11, 23, 29, …

Question 4 Gmat Quant Number Properties Tricks

Highest power of 3 dividing 50!?

⌊50/3⌋+⌊50/9⌋+⌊50/27⌋ = 16+5+1 = 22. So 3²². 1/3. 1.1·0.9 = 0.99 → 1% decrease. 1.5·0.5 = 0.75 → 25% decrease.

Question 5 Gmat Quant Number Properties Tricks

Number is divisible by both 4 and 6?

Need LCM(4,6)=12. Not 24 — common mistake. (n(n+1)/2)² — square of the sum. Odd × odd = odd; odd + odd = even. 5·(5/10) = 2.5 L.

Question 6 Gmat Quant Number Properties Tricks

Speed × time = distance. Time to cover 180 km at 60 km/h?

The last digit of a square is always 0, 1, 4, 5, 6, or 9. 3 hours. 20%. An integer divisible by the sum of its digits. E.g., 18: 1+8=9, 18/9=2 ✓.

Question 7 Gmat Quant Number Properties Tricks

Is 1458 divisible by 9?

12. (10% of 80 = 8, plus half = 4.) Number ≡ sum of digits (mod 9) — basis of divisibility tests. Digit sum 1+4+5+8 = 18 → yes. Also divisible by 3 and by 2. 4.

Question 8 Gmat Quant Number Properties Tricks

Divisibility by 2?

The last digit of a square is always 0, 1, 4, 5, 6, or 9. 36=2²·3² → φ=36·(1−½)·(1−⅓) = 36·½·⅔ = 12. 7≡2; 2⁴≡1; 600 mod 4 = 0 → 7⁶⁰⁰≡1. Last digit is even (0, 2, 4, 6, 8).

Question 9 Gmat Quant Number Properties Tricks

Alligation: 10 L of 30% acid, 40 L of 60% acid. Final concentration?

3628800. (300+2400)/50 = 54%. Same cycle as 3: 3,9,7,1. 79 mod 4 = 3 → last digit 7. 5050.

Question 10 Gmat Quant Number Properties Tricks

Are 18 and 35 coprime?

An integer divisible by the sum of its digits. E.g., 18: 1+8=9, 18/9=2 ✓. 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 ✓. Yes — share no prime factors, so GCD = 1 and LCM = 18·35 = 630. 5²=25≡−1; 5²²≡1; 5²³≡5. Remainder 5.

Question 11 Gmat Quant Number Properties Tricks

Geometric mean of 4 and 36?

1 − 0.8·0.9 = 0.28 → 28% off. √144 = 12. Arithmetic series with 10 terms, average 15.5 → 155. Always even (n(n−1)(n+1) is divisible by 2 and 3).

Question 12 Gmat Quant Number Properties Tricks

Recognize 0.1666…?

1/6. 1 − 0.8·0.9 = 0.28 → 28% off. 2520 = LCM(1,…,10). Suppose √2 = p/q in lowest terms → 2q² = p² → p even → p = 2k → q² = 2k² → q even — contradiction.

Question 13 Gmat Quant Number Properties Tricks

15 is what percent of 75?

Every integer > 1 has a unique prime factorization (up to order). 20%. Divisible by both 3 and 4. An integer of the form n² (n ≥ 0): 0, 1, 4, 9, 16, 25, …

Question 14 Gmat Quant Number Properties Tricks

Express 0.625 as a fraction?

12.5%. 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 ✓. 625/1000 = 5/8. Digit sum 1+4+5+8 = 18 → yes. Also divisible by 3 and by 2.

Question 15 Gmat Quant Number Properties Tricks

Rule of 72: doubling time at r% per year ≈ ?

(x−y)(x+y). 48=2⁴·3, 180=2²·3²·5 → 2²·3 = 12. 72/r years. E.g., 6% growth doubles in ~12 years. The last digit of a square is always 0, 1, 4, 5, 6, or 9.

Question 16 Gmat Quant Number Properties Tricks

Remainder when 7⁶⁰⁰ is divided by 5?

2/(1/2 + 1/8) = 16/5 = 3.2. 1/6. 7≡2; 2⁴≡1; 600 mod 4 = 0 → 7⁶⁰⁰≡1. Cycle 2,4,8,6 of length 4. 50 mod 4 = 2 → last digit 4.

Question 17 Gmat Quant Number Properties Tricks

Odd × odd = ?

Odd. 25/5 + 25/25 = 5 + 1 = 6. 3628800. (x−y)(x+y).

Question 18 Gmat Quant Number Properties Tricks

Last digit of 13⁷⁹?

2(1+2+…+20) = 2·210 = 420. No. 91 = 7 × 13. Same cycle as 3: 3,9,7,1. 79 mod 4 = 3 → last digit 7. 100·0.8·1.08 = $86.40.

Question 19 Gmat Quant Number Properties Tricks

Range of −3, 7, 12, 0, 5?

12 − (−3) = 15. 7·(1+2+…+100) = 7·5050 = 35350. Doesn't factor over integers; over complex: (x+iy)(x−iy). ⌊100/2⌋+⌊100/4⌋+⌊100/8⌋+⌊100/16⌋+⌊100/32⌋+⌊100/64⌋ = 50+25+12+6+3+1 = 97. So 2⁹⁷.

Question 20 Gmat Quant Number Properties Tricks

Can a² + b² be a perfect square?

Yes — Pythagorean triples. E.g., 3² + 4² = 5². Reads the same forwards and backwards. E.g., 121, 1331, 12321. 111111 = 111·1001 = (3·37)·(7·11·13). The last digit of a square is always 0, 1, 4, 5, 6, or 9.

Question 21 Gmat Quant Number Properties Tricks

C(10,3) (binomial coefficient)?

10!/(3!·7!) = 120. Whole numbers: 0, 1, 2, 3…
Integers: …, -2, -1, 0, 1, 2… Odd. Always 0 or 1.

Question 22 Gmat Quant Number Properties Tricks

Is 105 divisible by 7?

A prime of form 2^(2ⁿ)+1. Known: 3, 5, 17, 257, 65537. 50² = 2500. 10 − (2×5) = 0 → yes. (105 = 7 × 15.) Two numbers where each is the sum of proper divisors of the other. Smallest: (220, 284).

Question 23 Gmat Quant Number Properties Tricks

What is 15% of 80?

12. (10% of 80 = 8, plus half = 4.) 20% increase then 20% decrease ≠ original. 100 → 120 → 96. Always multiply: 1.2 × 0.8 = 0.96. 72/r years. E.g., 6% growth doubles in ~12 years. Always even (n(n−1)(n+1) is divisible by 2 and 3).

Question 24 Gmat Quant Number Properties Tricks

What does it mean for a number to be abundant?

20%. (300+2400)/50 = 54%. Last digit is even (0, 2, 4, 6, 8). Sum of proper divisors exceeds the number. E.g., 12: 1+2+3+4+6 = 16 > 12.

Question 25 Gmat Quant Number Properties Tricks

Same direction, 60 km/h and 40 km/h, starting 100 km apart. Time for faster to catch slower?

Even. 100·0.8·1.08 = $86.40. 100/20 = 5 hours. 33 terms, average 51 → 1683.

Question 26 Gmat Quant Number Properties Tricks

What is a Kaprekar number?

1/12+1/18 = 5/36 per day → 36/5 = 7.2 days. Ends in its own square. 5²=25, 6²=36, 25²=625, 76²=5776. 2·465 = 930. Also = 30·31 = 930. A number whose square splits into two parts summing to the original. E.g., 45²=2025 → 20+25=45.

Question 27 Gmat Quant Number Properties Tricks

Product of consecutive integers includes…?

48=2⁴·3, 180=2²·3²·5 → 2²·3 = 12. a = m²−n², b = 2mn, c = m²+n² (coprime m>n, opposite parity). Count factors of 5 (since 2s are plentiful): floor(n/5) + floor(n/25) + floor(n/125) + ... 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.

Question 28 Gmat Quant Number Properties Tricks

Sum of first n positive integers?

1/12+1/18 = 5/36 per day → 36/5 = 7.2 days. n(n+1)/2. 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). Only (2, 3).

Question 29 Gmat Quant Number Properties Tricks

Is 1234567 a perfect square?

Count factors of 5 (since 2s are plentiful): floor(n/5) + floor(n/25) + floor(n/125) + ... No — mod 4: 1234567 mod 4 = 3, but squares mod 4 are 0 or 1. x > 1: x² > x.
0 < x < 1: x² < x. Easy trap. 1/9.

Question 30 Gmat Quant Number Properties Tricks

Odd + odd = ?

The last digit of a square is always 0, 1, 4, 5, 6, or 9. The rightmost digit of an integer, equal to value mod 10. Even. Even (product of two consecutive integers).

Question 31 Gmat Quant Number Properties Tricks

What does it mean for a number to be perfect?

1.03¹⁰ ≈ 1.3439. Sum of proper divisors equals the number. E.g., 6, 28, 496, 8128. Congruent to 0 or 1 mod 4. Prime-factorize, add 1 to each exponent, multiply. 12 = 2² × 3¹ → (2+1)(1+1) = 6 divisors.

Question 32 Gmat Quant Number Properties Tricks

What is a repunit?

Even AND divisible by 3. 6 km/h. A number consisting only of the digit 1: 1, 11, 111, 1111, … (v²−u²)/v.

Question 33 Gmat Quant Number Properties Tricks

Sum of first 100 multiples of 7?

7·(1+2+…+100) = 7·5050 = 35350. 6. Among three consecutive integers, one is a multiple of 3 and at least one is even. 1/B = 1/8−1/12 = 1/24 → 24 days. A prime p such that 2p+1 is also prime. E.g., 2, 3, 5, 11, 23, 29, …

Question 34 Gmat Quant Number Properties Tricks

Divisibility by 125?

Last digit is even (0, 2, 4, 6, 8). 5²=25≡−1; 5²²≡1; 5²³≡5. Remainder 5. Last three digits divisible by 8. Last three digits are 000, 125, 250, 375, 500, 625, 750, or 875.

Question 35 Gmat Quant Number Properties Tricks

Last digit of 2⁵⁰?

Last digit is 0 or 5. Cycle 2,4,8,6 of length 4. 50 mod 4 = 2 → last digit 4. 3 hours. 12. (10% of 80 = 8, plus half = 4.)

Question 36 Gmat Quant Number Properties Tricks

Effective speed upstream (boat 10, current 4)?

10!/(3!·7!) = 120. 14 km/h. (a+b)/2 — also the median. 6 km/h.

Question 37 Gmat Quant Number Properties Tricks

What is a Fermat prime?

Cycle 3,9,7,1 of length 4. 100 mod 4 = 0 → last digit 1. 1/6−1/9 = 1/18 per min → 18 min. A prime of form 2^(2ⁿ)+1. Known: 3, 5, 17, 257, 65537. 64, 81, 100, 121, 144, 169, 196 — seven.

Question 38 Gmat Quant Number Properties Tricks

If x and y are integers and 2x + 3y is even, what's true?

y must be even (3y has same parity as y; for sum to be even, 3y must be even). 6. Among three consecutive integers, one is a multiple of 3 and at least one is even. 5^1=5, 5^2=25, 5^3=125, 5^4=625, 5^5=3125. Last digit always 5. ⌊log₁₀(50!)⌋ + 1 = 65.

Question 39 Gmat Quant Number Properties Tricks

Pattern of last two digits of n²?

7³≡1 (mod 9); 50 mod 3 = 2 → 7⁵⁰ ≡ 7² = 49 ≡ 4 (mod 9). Reads the same forwards and backwards. E.g., 121, 1331, 12321. 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. 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).

Question 40 Gmat Quant Number Properties Tricks

Average speed formula for two equal distances?

1.03¹⁰ ≈ 1.3439. No — mod 4: 1234567 mod 4 = 3, but squares mod 4 are 0 or 1. 2ab/(a+b) (harmonic mean), where a and b are the two speeds. Even (product of two consecutive integers).

Question 41 Gmat Quant Number Properties Tricks

Smallest Pythagorean triple with integer sides?

n(n+1)/2. Four: 53, 59, 61, 67. (51=3·17, 55=5·11, 57=3·19, 63=7·9, 65=5·13, 69=3·23.) (3, 4, 5). Last two digits divisible by 4. E.g., 1532 → 32 ÷ 4 = 8.

Question 42 Gmat Quant Number Properties Tricks

How many primes between 50 and 70?

10 − (2×5) = 0 → yes. (105 = 7 × 15.) Alternating sum: 1 − 4 + 3 = 0 → yes. (143 = 11 × 13.) 142857/999999 = 1/7. Four: 53, 59, 61, 67. (51=3·17, 55=5·11, 57=3·19, 63=7·9, 65=5·13, 69=3·23.)

Question 43 Gmat Quant Number Properties Tricks

What does it mean for a number to be deficient?

Digit sum 1+4+5+8 = 18 → yes. Also divisible by 3 and by 2. Sum of proper divisors is less than the number. E.g., 8: 1+2+4 = 7 < 8. 10 − (2×5) = 0 → yes. (105 = 7 × 15.) √144 = 12.

Question 44 Gmat Quant Number Properties Tricks

Divisibility by 6?

7·(1+2+…+100) = 7·5050 = 35350. Even AND divisible by 3. n(n+1)(2n+1)/6. Last two digits divisible by 4. E.g., 1532 → 32 ÷ 4 = 8.

Question 45 Gmat Quant Number Properties Tricks

What is a Mersenne prime?

7³≡1 (mod 9); 50 mod 3 = 2 → 7⁵⁰ ≡ 7² = 49 ≡ 4 (mod 9). 8: 2, 3, 5, 7, 11, 13, 17, 19. 12. (10% of 80 = 8, plus half = 4.) A prime of form 2ᵖ−1 with p prime. E.g., 3, 7, 31, 127, 8191.

Question 46 Gmat Quant Number Properties Tricks

Squares mod 4?

64, 81, 100, 121, 144, 169, 196 — seven. (n(n+1)/2)² — square of the sum. Always 0 or 1. No. 91 = 7 × 13.

Question 47 Gmat Quant Number Properties Tricks

Smallest positive integer divisible by every integer from 1 to 10?

12.5%. n(n+1)(2n+1)/6. An integer divisible by the sum of its digits. E.g., 18: 1+8=9, 18/9=2 ✓. 2520 = LCM(1,…,10).

Question 48 Gmat Quant Number Properties Tricks

Sum of squares: x² + y² factors over... ?

Yes — Pythagorean triples. E.g., 3² + 4² = 5². Even. Doesn't factor over integers; over complex: (x+iy)(x−iy). 360 = 2³ × 3² × 5 → (3+1)(2+1)(1+1) = 24.

Question 49 Gmat Quant Number Properties Tricks

Fundamental Theorem of Arithmetic?

A number whose square splits into two parts summing to the original. E.g., 45²=2025 → 20+25=45. Every integer > 1 has a unique prime factorization (up to order). 25/5 + 25/25 = 5 + 1 = 6. Two numbers where each is the sum of proper divisors of the other. Smallest: (220, 284).

Question 50 Gmat Quant Number Properties Tricks

Sum of first n odd numbers?

7≡2; 2⁴≡1; 600 mod 4 = 0 → 7⁶⁰⁰≡1. Congruent to 0 or 1 mod 4. 2. n².