What is the average (arithmetic mean) of the 5 consecutive integers from 24 to 28?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 26.
2
What is the sum of all positive divisors of 128 (including 128 itself)?
Medium
ExplanationThe divisors of 128 are [1, 2, 4, 8, 16, 32, 64, 128], which sum to 255.
3
How many prime numbers are there between 71 and 88, inclusive?
Medium
ExplanationThe primes between 71 and 88 are [71, 73, 79, 83], totaling 4.
4
What is the smallest number that must be subtracted from 1368 so that the result is exactly divisible by 37?
Medium
Explanation1368 divided by 37 leaves remainder 36, so subtracting 36 makes it exactly divisible by 37.
5
What is the greatest common divisor of 8, 36, and 24?
Medium
ExplanationThe largest integer dividing all of 8, 36, and 24 evenly is 4.
6
9725 is exactly divisible by which of the following numbers?
Medium
Explanation9725 ÷ 5 = 1945, a whole number, so 5 divides 9725 exactly.
7
What is the highest power of 13 that divides 76! (76 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 13 in 76! is floor(76/13) + floor(76/169) + ... = 5.
8
What is the smallest positive integer that leaves a remainder of 4 when divided by 10 and a remainder of 1 when divided by 7?
Medium
ExplanationSince both remainders are exactly 6 less than their divisors, the answer is LCM(10, 7) − 6 = 70 − 6 = 64.
9
What is the greatest common divisor of 108, 48, and 120?
Medium
ExplanationThe largest integer dividing all of 108, 48, and 120 evenly is 12.
10
An even number raised to any positive integer power is always
Medium
ExplanationThis follows directly from the basic parity rules of odd and even integers.
— Sponsored —
11
Three warning lights flash every 20, 14, 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, 14, 4) = 140 seconds.
12
What is the highest power of 11 that divides 61! (61 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 11 in 61! is floor(61/11) + floor(61/121) + ... = 5.
13
What is the sum of 3 consecutive integers starting from 28?
Medium
ExplanationThe integers are 28 through 30; their sum is 87.
14
Three warning lights flash every 9, 14, and 15 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(9, 14, 15) = 630 seconds.
15
Two buses leave a station together. One returns to the station every 32 minutes and the other every 17 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(32, 17) = 544 minutes.
16
If two positive integers are 27 and 17, what is the product of their LCM and GCD?
Medium
ExplanationFor any two positive integers, LCM × GCD = product of the numbers = 27 × 17 = 459.
17
Three warning lights flash every 12, 14, and 20 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(12, 14, 20) = 420 seconds.
18
What is the smallest number that must be subtracted from 9665 so that the result is exactly divisible by 44?
Medium
Explanation9665 divided by 44 leaves remainder 29, so subtracting 29 makes it exactly divisible by 44.
19
What is the sum of 8 consecutive integers starting from 13?
Medium
ExplanationThe integers are 13 through 20; their sum is 132.
20
Are 81 and 27 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(81, 27) = 27, so they are not relatively prime, since they share the common factor 27.
— Sponsored —
21
What is the smallest positive integer that is divisible by all of 20, 24, and 18?
Medium
ExplanationThe smallest such number is LCM(20, 24, 18) = 360.
22
Is 99 a prime number?
Medium
Explanation99 is divisible by at least one number between 2 and 98, so it is not prime.
23
What is the average (arithmetic mean) of the 7 consecutive integers from 60 to 66?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 63.
24
How many positive factors (divisors) does 374 have?
Medium
Explanation374 = 2^1 × 11^1 × 17^1. The number of divisors is the product of (exponent+1) for each prime factor = 8.
25
How many prime numbers are there between 5 and 30, inclusive?
Medium
ExplanationThe primes between 5 and 30 are [5, 7, 11, 13, 17, 19, 23, 29], totaling 8.
26
3894 is exactly divisible by which of the following numbers?
Medium
Explanation3894 ÷ 2 = 1947, a whole number, so 2 divides 3894 exactly.
27
What is the smallest number that must be subtracted from 6915 so that the result is exactly divisible by 10?
Medium
Explanation6915 divided by 10 leaves remainder 5, so subtracting 5 makes it exactly divisible by 10.
28
How many positive factors (divisors) does 153 have?
Medium
Explanation153 = 3^2 × 17^1. The number of divisors is the product of (exponent+1) for each prime factor = 6.
29
How many trailing zeros are in 37! (37 factorial)?
Medium
ExplanationTrailing zeros come from factors of 5 in 37!. Counting multiples of 5, 25, 125, ... gives 8 trailing zeros.
30
Does the fraction 4/138 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 4/138 = 2/69. 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).
— Sponsored —
31
What is the highest power of 2 that divides 67! (67 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 2 in 67! is floor(67/2) + floor(67/4) + ... = 64.
32
If an integer n leaves a remainder of 1 when divided by 12, what is the remainder when n^2 is divided by 12?
Medium
Explanationn = 12q + 1 for some integer q, so n^2 ≡ 1^2 = 1 (mod 12), which leaves remainder 1 when divided by 12.
33
What is the smallest positive integer that leaves a remainder of 2 when divided by 4 and a remainder of 11 when divided by 13?
Medium
ExplanationSince both remainders are exactly 2 less than their divisors, the answer is LCM(4, 13) − 2 = 52 − 2 = 50.
34
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.
35
How many distinct prime factors does 417 have?
Medium
ExplanationThe prime factorization of 417 is 3^1 × 139^1, which includes 2 distinct prime(s).
36
What is the sum of 3 consecutive integers starting from 26?
Medium
ExplanationThe integers are 26 through 28; their sum is 81.
37
What is the smallest number that must be subtracted from 6814 so that the result is exactly divisible by 28?
Medium
Explanation6814 divided by 28 leaves remainder 10, so subtracting 10 makes it exactly divisible by 28.
38
Does the fraction 30/187 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 30/187 = 30/187. 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).
39
Is 323 a prime number?
Medium
Explanation323 is divisible by at least one number between 2 and 322, so it is not prime.
40
What is the sum of all positive divisors of 119 (including 119 itself)?
Medium
ExplanationThe divisors of 119 are [1, 7, 17, 119], which sum to 144.
— Sponsored —
41
What is the average (arithmetic mean) of the 9 consecutive integers from 39 to 47?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 43.
42
How many multiples of 4 are there between 100 and 381, inclusive?
Medium
ExplanationThe multiples of 4 in this range run from 100 to 380, giving 71 multiples.
43
The digits of 1517 sum to 14. Based on this, is 1517 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.
44
What is the average (arithmetic mean) of the 7 consecutive integers from 10 to 16?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 13.
45
What is the least common multiple (LCM) of 40 and 19?
Medium
ExplanationLCM(40, 19) = 760, the smallest number divisible by both 40 and 19.
46
What is the average (arithmetic mean) of the 7 consecutive integers from 8 to 14?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 11.
47
Two buses leave a station together. One returns to the station every 11 minutes and the other every 13 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(11, 13) = 143 minutes.
48
How many multiples of 11 are there between 2 and 255, inclusive?
Medium
ExplanationThe multiples of 11 in this range run from 11 to 253, giving 23 multiples.
49
How many positive factors (divisors) does 184 have?
Medium
Explanation184 = 2^3 × 23^1. The number of divisors is the product of (exponent+1) for each prime factor = 8.
50
How many prime numbers are there between 21 and 44, inclusive?
Medium
ExplanationThe primes between 21 and 44 are [23, 29, 31, 37, 41, 43], totaling 6.
— Sponsored —
51
Two buses leave a station together. One returns to the station every 15 minutes and the other every 32 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(15, 32) = 480 minutes.
52
How many prime numbers are there between 20 and 36, inclusive?
Medium
ExplanationThe primes between 20 and 36 are [23, 29, 31], totaling 3.
53
What is the smallest positive integer that is divisible by all of 21, 6, and 22?
Medium
ExplanationThe smallest such number is LCM(21, 6, 22) = 462.
54
The digits of 9455 sum to 23. Based on this, is 9455 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.
55
Is 117 a prime number?
Medium
Explanation117 is divisible by at least one number between 2 and 116, so it is not prime.
56
What is the greatest common divisor (GCD) of 56 and 52?
Medium
ExplanationThe largest integer dividing both 56 and 52 evenly is 4.
57
Are 66 and 55 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(66, 55) = 11, so they are not relatively prime, since they share the common factor 11.
58
What is the smallest positive number that must be added to 397 so that the sum is exactly divisible by 45?
Medium
Explanation397 divided by 45 leaves remainder 37, so adding 45 − 37 = 8 makes it exactly divisible by 45.
59
What is the greatest common divisor of 60, 100, and 70?
Medium
ExplanationThe largest integer dividing all of 60, 100, and 70 evenly is 10.
60
If two positive integers are 27 and 25, what is the product of their LCM and GCD?
Medium
ExplanationFor any two positive integers, LCM × GCD = product of the numbers = 27 × 25 = 675.
— Sponsored —
61
How many prime numbers are there between 23 and 38, inclusive?
Medium
ExplanationThe primes between 23 and 38 are [23, 29, 31, 37], totaling 4.
62
If an integer n leaves a remainder of 4 when divided by 6, what is the remainder when n^2 is divided by 6?
Medium
Explanationn = 6q + 4 for some integer q, so n^2 ≡ 4^2 = 16 (mod 6), which leaves remainder 4 when divided by 6.
63
How many positive factors (divisors) does 206 have?
Medium
Explanation206 = 2^1 × 103^1. The number of divisors is the product of (exponent+1) for each prime factor = 4.
64
How many prime numbers are there between 41 and 61, inclusive?
Medium
ExplanationThe primes between 41 and 61 are [41, 43, 47, 53, 59, 61], totaling 6.
65
Three warning lights flash every 19, 20, and 14 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, 20, 14) = 2660 seconds.
66
What is the average (arithmetic mean) of the 3 consecutive integers from 18 to 20?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 19.
67
What is the greatest common divisor of 169, 26, and 39?
Medium
ExplanationThe largest integer dividing all of 169, 26, and 39 evenly is 13.
68
Does the fraction 10/46 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 10/46 = 5/23. 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).
69
24^2 − 9^2 is exactly divisible by which of the following?
Medium
Explanation24^2 − 9^2 = (24 − 9)(24 + 9) = 15 × 33, so it is always divisible by 15 (and by 33).
70
What is the smallest positive integer that is divisible by all of 5, 16, and 23?
Medium
ExplanationThe smallest such number is LCM(5, 16, 23) = 1840.
— Sponsored —
71
How many positive factors (divisors) does 201 have?
Medium
Explanation201 = 3^1 × 67^1. The number of divisors is the product of (exponent+1) for each prime factor = 4.
72
If two positive integers are 19 and 29, what is the product of their LCM and GCD?
Medium
ExplanationFor any two positive integers, LCM × GCD = product of the numbers = 19 × 29 = 551.
73
9611 is exactly divisible by which of the following numbers?
Medium
Explanation9611 ÷ 7 = 1373, a whole number, so 7 divides 9611 exactly.
74
Is 223 a prime number?
Medium
Explanation223 is not divisible by any integer other than 1 and itself, so it is prime.
75
What is the sum of 8 consecutive integers starting from 28?
Medium
ExplanationThe integers are 28 through 35; their sum is 252.
76
What is the smallest positive integer that leaves a remainder of 3 when divided by 6 and a remainder of 4 when divided by 7?
Medium
ExplanationSince both remainders are exactly 3 less than their divisors, the answer is LCM(6, 7) − 3 = 42 − 3 = 39.
77
If an integer n leaves a remainder of 10 when divided by 14, what is the remainder when n^2 is divided by 14?
Medium
Explanationn = 14q + 10 for some integer q, so n^2 ≡ 10^2 = 100 (mod 14), which leaves remainder 2 when divided by 14.
78
Three warning lights flash every 4, 15, and 16 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(4, 15, 16) = 240 seconds.
79
If an integer n leaves a remainder of 13 when divided by 15, what is the remainder when n^2 is divided by 15?
Medium
Explanationn = 15q + 13 for some integer q, so n^2 ≡ 13^2 = 169 (mod 15), which leaves remainder 4 when divided by 15.
80
What is the sum of 4 consecutive integers starting from 41?
Medium
ExplanationThe integers are 41 through 44; their sum is 170.
— Sponsored —
81
An odd number raised to any positive integer power is always
Medium
ExplanationThis follows directly from the basic parity rules of odd and even integers.
82
What is the smallest positive number that must be added to 6351 so that the sum is exactly divisible by 6?
Medium
Explanation6351 divided by 6 leaves remainder 3, so adding 6 − 3 = 3 makes it exactly divisible by 6.
83
How many digits does 3^46 have?
Medium
ExplanationThe number of digits of 3^46 is floor(46 × log10(3)) + 1 = 22.
84
What is the least common multiple (LCM) of 7 and 25?
Medium
ExplanationLCM(7, 25) = 175, the smallest number divisible by both 7 and 25.
85
The digits of 6316 sum to 16. Based on this, is 6316 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 16, which is not divisible by 3.
86
Three warning lights flash every 18, 10, and 16 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(18, 10, 16) = 720 seconds.
87
Three warning lights flash every 16, 6, and 8 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, 6, 8) = 48 seconds.
88
What is the greatest common divisor of 30, 135, and 75?
Medium
ExplanationThe largest integer dividing all of 30, 135, and 75 evenly is 15.
89
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.
90
Is 36779 divisible by 11?
Medium
ExplanationUsing the alternating digit-sum rule for 11, the result is 6, which is not a multiple of 11, so the answer is No.
— Sponsored —
91
Are 86 and 31 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(86, 31) = 1, so they are relatively prime.
92
What is the sum of 6 consecutive even integers starting from 40?
Medium
ExplanationThe 6 consecutive even integers are [40, 42, 44, 46, 48, 50]; their sum is 270.
93
What is the smallest number that must be subtracted from 9244 so that the result is exactly divisible by 16?
Medium
Explanation9244 divided by 16 leaves remainder 12, so subtracting 12 makes it exactly divisible by 16.
94
What is the average (arithmetic mean) of the 7 consecutive integers from 25 to 31?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 28.
95
How many multiples of 9 are there between 88 and 355, inclusive?
Medium
ExplanationThe multiples of 9 in this range run from 90 to 351, giving 30 multiples.
96
Are 30 and 33 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(30, 33) = 3, so they are not relatively prime, since they share the common factor 3.
97
Two buses leave a station together. One returns to the station every 22 minutes and the other every 30 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(22, 30) = 330 minutes.
98
If an integer n leaves a remainder of 6 when divided by 8, what is the remainder when 5n is divided by 8?
Medium
Explanation5n ≡ 5 × 6 = 30 (mod 8), which leaves remainder 6 when divided by 8.
99
Which of the following numbers is composite?
Medium
Explanation225 is divisible by 3 in addition to 1 and itself, so it is composite.
100
Are 72 and 17 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(72, 17) = 1, so they are relatively prime.