What is the average (arithmetic mean) of the 5 consecutive integers from 75 to 79?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 77.
2
Two buses leave a station together. One returns to the station every 13 minutes and the other every 40 minutes. After how many minutes will both buses be at the station together again?
Medium
ExplanationBoth buses will be at the station together again after LCM(13, 40) = 520 minutes.
3
What is the smallest number that must be subtracted from 3539 so that the result is exactly divisible by 31?
Medium
Explanation3539 divided by 31 leaves remainder 5, so subtracting 5 makes it exactly divisible by 31.
4
40^2 = 1600. How many positive divisors does 1600 have?
Medium
Explanation1600 = 2^6 × 5^2, giving 21 divisors — always an odd count for a perfect square, since every divisor pairs with another except the square root.
5
If an integer n leaves a remainder of 7 when divided by 14, what is the remainder when n^2 is divided by 14?
Medium
Explanationn = 14q + 7 for some integer q, so n^2 ≡ 7^2 = 49 (mod 14), which leaves remainder 7 when divided by 14.
6
What is the highest power of 13 that divides 60! (60 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 13 in 60! is floor(60/13) + floor(60/169) + ... = 4.
7
How many multiples of 9 are there between 156 and 426, inclusive?
Medium
ExplanationThe multiples of 9 in this range run from 162 to 423, giving 30 multiples.
8
How many distinct prime factors does 45 have?
Medium
ExplanationThe prime factorization of 45 is 3^2 × 5^1, which includes 2 distinct prime(s).
9
How many positive factors (divisors) does 333 have?
Medium
Explanation333 = 3^2 × 37^1. The number of divisors is the product of (exponent+1) for each prime factor = 6.
10
If two positive integers are 14 and 24, what is the product of their LCM and GCD?
Medium
ExplanationFor any two positive integers, LCM × GCD = product of the numbers = 14 × 24 = 336.
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11
What is the sum of 3 consecutive integers starting from 29?
Medium
ExplanationThe integers are 29 through 31; their sum is 90.
12
The digits of 2032 sum to 7. Based on this, is 2032 divisible by 3?
Medium
ExplanationA number is divisible by 3 if and only if its digit sum is divisible by 3. Here the digit sum is 7, which is not divisible by 3.
13
How many positive factors (divisors) does 254 have?
Medium
Explanation254 = 2^1 × 127^1. The number of divisors is the product of (exponent+1) for each prime factor = 4.
14
What is the highest power of 13 that divides 27! (27 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 13 in 27! is floor(27/13) + floor(27/169) + ... = 2.
15
If two positive integers are 24 and 26, what is the product of their LCM and GCD?
Medium
ExplanationFor any two positive integers, LCM × GCD = product of the numbers = 24 × 26 = 624.
16
What is the smallest positive integer that is divisible by all of 4, 6, and 19?
Medium
ExplanationThe smallest such number is LCM(4, 6, 19) = 228.
17
What is the smallest positive number that must be added to 4266 so that the sum is exactly divisible by 36?
Medium
Explanation4266 divided by 36 leaves remainder 18, so adding 36 − 18 = 18 makes it exactly divisible by 36.
18
What is the smallest positive number that must be added to 1165 so that the sum is exactly divisible by 27?
Medium
Explanation1165 divided by 27 leaves remainder 4, so adding 27 − 4 = 23 makes it exactly divisible by 27.
19
What is the units digit of 5^20?
Medium
ExplanationUnits digits of powers of 5 repeat in a cycle of length 1. Using 20 mod 1, the units digit is 5.
20
Is 76877 divisible by 11?
Medium
ExplanationUsing the alternating digit-sum rule for 11, the result is 9, which is not a multiple of 11, so the answer is No.
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21
What is the sum of 6 consecutive odd integers starting from 1?
Medium
ExplanationThe 6 consecutive odd integers are [1, 3, 5, 7, 9, 11]; their sum is 36.
22
What is the smallest positive integer that is divisible by all of 18, 16, and 9?
Medium
ExplanationThe smallest such number is LCM(18, 16, 9) = 144.
23
What is the smallest positive number that must be added to 7892 so that the sum is exactly divisible by 16?
Medium
Explanation7892 divided by 16 leaves remainder 4, so adding 16 − 4 = 12 makes it exactly divisible by 16.
24
Three warning lights flash every 20, 5, and 4 seconds respectively. If they all flash together at the same instant, after how many seconds will they next flash together?
Medium
ExplanationThey will next flash together after LCM(20, 5, 4) = 20 seconds.
25
If an integer n leaves a remainder of 4 when divided by 11, what is the remainder when 9n is divided by 11?
Medium
Explanation9n ≡ 9 × 4 = 36 (mod 11), which leaves remainder 3 when divided by 11.
26
40^2 − 35^2 is exactly divisible by which of the following?
Medium
Explanation40^2 − 35^2 = (40 − 35)(40 + 35) = 5 × 75, so it is always divisible by 5 (and by 75).
27
Does the fraction 27/107 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 27/107 = 27/107. A fraction terminates only if its denominator (in lowest terms) has no prime factors other than 2 and 5. Here that is not the case, so it repeats (does not terminate).
28
How many positive factors (divisors) does 99 have?
Medium
Explanation99 = 3^2 × 11^1. The number of divisors is the product of (exponent+1) for each prime factor = 6.
29
What is the smallest number that must be subtracted from 2484 so that the result is exactly divisible by 38?
Medium
Explanation2484 divided by 38 leaves remainder 14, so subtracting 14 makes it exactly divisible by 38.
30
Does the fraction 26/396 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 26/396 = 13/198. A fraction terminates only if its denominator (in lowest terms) has no prime factors other than 2 and 5. Here that is not the case, so it repeats (does not terminate).
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31
Does the fraction 27/499 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 27/499 = 27/499. A fraction terminates only if its denominator (in lowest terms) has no prime factors other than 2 and 5. Here that is not the case, so it repeats (does not terminate).
32
What is the sum of 8 consecutive integers starting from 13?
Medium
ExplanationThe integers are 13 through 20; their sum is 132.
33
What is the highest power of 2 that divides 60! (60 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 2 in 60! is floor(60/2) + floor(60/4) + ... = 56.
34
What is the remainder when 406 is divided by 5?
Medium
Explanation406 = 5 × 81 + 1, so the remainder is 1.
35
What is the average (arithmetic mean) of the 3 consecutive integers from 78 to 80?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 79.
36
If two positive integers are 4 and 14, what is the product of their LCM and GCD?
Medium
ExplanationFor any two positive integers, LCM × GCD = product of the numbers = 4 × 14 = 56.
37
1090 is NOT divisible by which of the following numbers?
Medium
Explanation1090 ÷ 11 leaves a remainder of 1, so it does not divide evenly.
38
How many prime numbers are there between 62 and 91, inclusive?
Medium
ExplanationThe primes between 62 and 91 are [67, 71, 73, 79, 83, 89], totaling 6.
39
What is the least common multiple (LCM) of 24 and 27?
Medium
ExplanationLCM(24, 27) = 216, the smallest number divisible by both 24 and 27.
40
What is the least common multiple (LCM) of 15 and 32?
Medium
ExplanationLCM(15, 32) = 480, the smallest number divisible by both 15 and 32.
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41
How many distinct prime factors does 156 have?
Medium
ExplanationThe prime factorization of 156 is 2^2 × 3^1 × 13^1, which includes 3 distinct prime(s).
42
What is the units digit of 2^19?
Medium
ExplanationUnits digits of powers of 2 repeat in a cycle of length 4. Using 19 mod 4, the units digit is 8.
43
What is the greatest common divisor (GCD) of 42 and 112?
Medium
ExplanationThe largest integer dividing both 42 and 112 evenly is 14.
44
What is the greatest common divisor of 36, 63, and 108?
Medium
ExplanationThe largest integer dividing all of 36, 63, and 108 evenly is 9.
45
9774 is NOT divisible by which of the following numbers?
Medium
Explanation9774 ÷ 4 leaves a remainder of 2, so it does not divide evenly.
46
What is the average (arithmetic mean) of the 9 consecutive integers from 48 to 56?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 52.
47
What is the sum of all positive divisors of 24 (including 24 itself)?
Medium
ExplanationThe divisors of 24 are [1, 2, 3, 4, 6, 8, 12, 24], which sum to 60.
48
What is the highest power of 11 that divides 49! (49 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 11 in 49! is floor(49/11) + floor(49/121) + ... = 4.
49
If two positive integers are 22 and 14, what is the product of their LCM and GCD?
Medium
ExplanationFor any two positive integers, LCM × GCD = product of the numbers = 22 × 14 = 308.
50
8768 is exactly divisible by which of the following numbers?
Medium
Explanation8768 ÷ 2 = 4384, a whole number, so 2 divides 8768 exactly.
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51
How many positive factors (divisors) does 338 have?
Medium
Explanation338 = 2^1 × 13^2. The number of divisors is the product of (exponent+1) for each prime factor = 6.
52
What is the greatest common divisor (GCD) of 15 and 36?
Medium
ExplanationThe largest integer dividing both 15 and 36 evenly is 3.
53
Two buses leave a station together. One returns to the station every 19 minutes and the other every 23 minutes. After how many minutes will both buses be at the station together again?
Medium
ExplanationBoth buses will be at the station together again after LCM(19, 23) = 437 minutes.
54
What is the smallest number that must be subtracted from 4401 so that the result is exactly divisible by 43?
Medium
Explanation4401 divided by 43 leaves remainder 15, so subtracting 15 makes it exactly divisible by 43.
55
The digits of 5108 sum to 14. Based on this, is 5108 divisible by 3?
Medium
ExplanationA number is divisible by 3 if and only if its digit sum is divisible by 3. Here the digit sum is 14, which is not divisible by 3.
56
Does the fraction 12/345 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 12/345 = 4/115. A fraction terminates only if its denominator (in lowest terms) has no prime factors other than 2 and 5. Here that is not the case, so it repeats (does not terminate).
57
What is the smallest positive integer that is divisible by all of 8, 13, and 17?
Medium
ExplanationThe smallest such number is LCM(8, 13, 17) = 1768.
58
What is the smallest number that must be subtracted from 9641 so that the result is exactly divisible by 45?
Medium
Explanation9641 divided by 45 leaves remainder 11, so subtracting 11 makes it exactly divisible by 45.
59
Is 293 a prime number?
Medium
Explanation293 is not divisible by any integer other than 1 and itself, so it is prime.
60
What is the greatest common divisor (GCD) of 77 and 91?
Medium
ExplanationThe largest integer dividing both 77 and 91 evenly is 7.
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61
What is the least common multiple (LCM) of 21 and 9?
Medium
ExplanationLCM(21, 9) = 63, the smallest number divisible by both 21 and 9.
62
What is the units digit of 3^55?
Medium
ExplanationUnits digits of powers of 3 repeat in a cycle of length 4. Using 55 mod 4, the units digit is 7.
63
If an integer n leaves a remainder of 5 when divided by 9, what is the remainder when 8n is divided by 9?
Medium
Explanation8n ≡ 8 × 5 = 40 (mod 9), which leaves remainder 4 when divided by 9.
64
What is the sum of all positive divisors of 100 (including 100 itself)?
Medium
ExplanationThe divisors of 100 are [1, 2, 4, 5, 10, 20, 25, 50, 100], which sum to 217.
65
Two buses leave a station together. One returns to the station every 24 minutes and the other every 15 minutes. After how many minutes will both buses be at the station together again?
Medium
ExplanationBoth buses will be at the station together again after LCM(24, 15) = 120 minutes.
66
The digits of 2678 sum to 23. Based on this, is 2678 divisible by 3?
Medium
ExplanationA number is divisible by 3 if and only if its digit sum is divisible by 3. Here the digit sum is 23, which is not divisible by 3.
67
What is the highest power of 3 that divides 20! (20 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 3 in 20! is floor(20/3) + floor(20/9) + ... = 8.
68
Is 7446 divisible by 11?
Medium
ExplanationUsing the alternating digit-sum rule for 11, the result is -1, which is not a multiple of 11, so the answer is No.
69
The digits of 5377 sum to 22. Based on this, is 5377 divisible by 3?
Medium
ExplanationA number is divisible by 3 if and only if its digit sum is divisible by 3. Here the digit sum is 22, which is not divisible by 3.
70
Is 72512 divisible by 11?
Medium
ExplanationUsing the alternating digit-sum rule for 11, the result is 11, which is a multiple of 11, so the answer is Yes.
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71
What is the smallest number that must be subtracted from 5618 so that the result is exactly divisible by 44?
Medium
Explanation5618 divided by 44 leaves remainder 30, so subtracting 30 makes it exactly divisible by 44.
72
What is the smallest positive number that must be added to 1323 so that the sum is exactly divisible by 8?
Medium
Explanation1323 divided by 8 leaves remainder 3, so adding 8 − 3 = 5 makes it exactly divisible by 8.
73
Are 36 and 66 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(36, 66) = 6, so they are not relatively prime, since they share the common factor 6.
74
What is the sum of 4 consecutive integers starting from 33?
Medium
ExplanationThe integers are 33 through 36; their sum is 138.
75
What is the greatest common divisor of 52, 39, and 104?
Medium
ExplanationThe largest integer dividing all of 52, 39, and 104 evenly is 13.
76
What is the smallest positive integer that is divisible by all of 22, 20, and 12?
Medium
ExplanationThe smallest such number is LCM(22, 20, 12) = 660.
77
What is the units digit of 4^15?
Medium
ExplanationUnits digits of powers of 4 repeat in a cycle of length 2. Using 15 mod 2, the units digit is 4.
78
22^2 = 484. How many positive divisors does 484 have?
Medium
Explanation484 = 2^2 × 11^2, giving 9 divisors — always an odd count for a perfect square, since every divisor pairs with another except the square root.
79
How many digits does 3^43 have?
Medium
ExplanationThe number of digits of 3^43 is floor(43 × log10(3)) + 1 = 21.
80
Is 495 a prime number?
Medium
Explanation495 is divisible by at least one number between 2 and 494, so it is not prime.
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81
Two buses leave a station together. One returns to the station every 7 minutes and the other every 34 minutes. After how many minutes will both buses be at the station together again?
Medium
ExplanationBoth buses will be at the station together again after LCM(7, 34) = 238 minutes.
82
What is the average (arithmetic mean) of the 5 consecutive integers from 53 to 57?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 55.
83
What is the average (arithmetic mean) of the 7 consecutive integers from 80 to 86?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 83.
84
What is the smallest positive integer that leaves a remainder of 1 when divided by 13 and a remainder of 2 when divided by 14?
Medium
ExplanationSince both remainders are exactly 12 less than their divisors, the answer is LCM(13, 14) − 12 = 182 − 12 = 170.
85
Is 103 a prime number?
Medium
Explanation103 is not divisible by any integer other than 1 and itself, so it is prime.
86
What is the remainder when 55 is divided by 13?
Medium
Explanation55 = 13 × 4 + 3, so the remainder is 3.
87
If an integer n leaves a remainder of 6 when divided by 11, what is the remainder when (n + 17) is divided by 11?
Medium
Explanation(n + 17) ≡ (6 + 17) = 23 (mod 11), which leaves remainder 1 when divided by 11.
88
If an integer n leaves a remainder of 3 when divided by 11, what is the remainder when n^2 is divided by 11?
Medium
Explanationn = 11q + 3 for some integer q, so n^2 ≡ 3^2 = 9 (mod 11), which leaves remainder 9 when divided by 11.
89
How many multiples of 8 are there between 157 and 348, inclusive?
Medium
ExplanationThe multiples of 8 in this range run from 160 to 344, giving 24 multiples.
90
What is the greatest common divisor of 54, 12, and 30?
Medium
ExplanationThe largest integer dividing all of 54, 12, and 30 evenly is 6.
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91
Three warning lights flash every 16, 11, and 12 seconds respectively. If they all flash together at the same instant, after how many seconds will they next flash together?
Medium
ExplanationThey will next flash together after LCM(16, 11, 12) = 528 seconds.
92
Are 43 and 98 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(43, 98) = 1, so they are relatively prime.
93
What is the units digit of 6^40?
Medium
ExplanationUnits digits of powers of 6 repeat in a cycle of length 1. Using 40 mod 1, the units digit is 6.
94
How many multiples of 15 are there between 137 and 395, inclusive?
Medium
ExplanationThe multiples of 15 in this range run from 150 to 390, giving 17 multiples.
95
What is the smallest positive integer that is divisible by all of 4, 12, and 7?
Medium
ExplanationThe smallest such number is LCM(4, 12, 7) = 84.
96
Is 17616 divisible by 11?
Medium
ExplanationUsing the alternating digit-sum rule for 11, the result is 5, which is not a multiple of 11, so the answer is No.
97
What is the average (arithmetic mean) of the 9 consecutive integers from 55 to 63?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 59.
98
What is the units digit of 4^51?
Medium
ExplanationUnits digits of powers of 4 repeat in a cycle of length 2. Using 51 mod 2, the units digit is 4.
99
Three warning lights flash every 19, 6, and 9 seconds respectively. If they all flash together at the same instant, after how many seconds will they next flash together?
Medium
ExplanationThey will next flash together after LCM(19, 6, 9) = 342 seconds.
100
What is the remainder when 709 is divided by 10?
Medium
Explanation709 = 10 × 70 + 9, so the remainder is 9.