What is the smallest number that must be subtracted from 4794 so that the result is exactly divisible by 35?
Medium
Explanation4794 divided by 35 leaves remainder 34, so subtracting 34 makes it exactly divisible by 35.
2
What is the smallest number that must be subtracted from 9987 so that the result is exactly divisible by 20?
Medium
Explanation9987 divided by 20 leaves remainder 7, so subtracting 7 makes it exactly divisible by 20.
3
20^2 = 400. How many positive divisors does 400 have?
Medium
Explanation400 = 2^4 × 5^2, giving 15 divisors — always an odd count for a perfect square, since every divisor pairs with another except the square root.
4
How many distinct prime factors does 596 have?
Medium
ExplanationThe prime factorization of 596 is 2^2 × 149^1, which includes 2 distinct prime(s).
5
Are 89 and 77 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(89, 77) = 1, so they are relatively prime.
6
What is the highest power of 2 that divides 23! (23 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 2 in 23! is floor(23/2) + floor(23/4) + ... = 19.
7
Two buses leave a station together. One returns to the station every 26 minutes and the other every 38 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(26, 38) = 494 minutes.
8
What is the least common multiple (LCM) of 18 and 37?
Medium
ExplanationLCM(18, 37) = 666, the smallest number divisible by both 18 and 37.
9
What is the units digit of 5^37?
Medium
ExplanationUnits digits of powers of 5 repeat in a cycle of length 1. Using 37 mod 1, the units digit is 5.
10
What is the sum of 4 consecutive odd integers starting from 41?
Medium
ExplanationThe 4 consecutive odd integers are [41, 43, 45, 47]; their sum is 176.
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11
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.
12
21^2 − 10^2 is exactly divisible by which of the following?
Medium
Explanation21^2 − 10^2 = (21 − 10)(21 + 10) = 11 × 31, so it is always divisible by 11 (and by 31).
13
What is the sum of 3 consecutive integers starting from 2?
Medium
ExplanationThe integers are 2 through 4; their sum is 9.
14
Are 54 and 42 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(54, 42) = 6, so they are not relatively prime, since they share the common factor 6.
15
What is the smallest positive integer that is divisible by all of 8, 7, and 11?
Medium
ExplanationThe smallest such number is LCM(8, 7, 11) = 616.
16
What is the least common multiple (LCM) of 21 and 26?
Medium
ExplanationLCM(21, 26) = 546, the smallest number divisible by both 21 and 26.
17
How many prime numbers are there between 68 and 86, inclusive?
Medium
ExplanationThe primes between 68 and 86 are [71, 73, 79, 83], totaling 4.
18
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.
19
28^2 − 8^2 is exactly divisible by which of the following?
Medium
Explanation28^2 − 8^2 = (28 − 8)(28 + 8) = 20 × 36, so it is always divisible by 20 (and by 36).
20
What is the remainder when 477 is divided by 6?
Medium
Explanation477 = 6 × 79 + 3, so the remainder is 3.
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21
What is the greatest common divisor of 91, 70, and 63?
Medium
ExplanationThe largest integer dividing all of 91, 70, and 63 evenly is 7.
22
What is the smallest positive integer that is divisible by all of 21, 13, and 7?
Medium
ExplanationThe smallest such number is LCM(21, 13, 7) = 273.
23
What is the sum of 8 consecutive integers starting from 8?
Medium
ExplanationThe integers are 8 through 15; their sum is 92.
24
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.
25
What is the greatest common divisor (GCD) of 154 and 143?
Medium
ExplanationThe largest integer dividing both 154 and 143 evenly is 11.
26
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.
27
What is the highest power of 13 that divides 26! (26 factorial) exactly?
Medium
ExplanationUsing Legendre's formula, the exponent of 13 in 26! is floor(26/13) + floor(26/169) + ... = 2.
28
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.
29
5673 is exactly divisible by which of the following numbers?
Medium
Explanation5673 ÷ 3 = 1891, a whole number, so 3 divides 5673 exactly.
30
Are 27 and 9 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(27, 9) = 9, so they are not relatively prime, since they share the common factor 9.
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31
What is the smallest positive number that must be added to 1842 so that the sum is exactly divisible by 13?
Medium
Explanation1842 divided by 13 leaves remainder 9, so adding 13 − 9 = 4 makes it exactly divisible by 13.
32
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.
33
What is the units digit of 2^34?
Medium
ExplanationUnits digits of powers of 2 repeat in a cycle of length 4. Using 34 mod 4, the units digit is 4.
34
Three warning lights flash every 20, 17, and 5 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, 17, 5) = 340 seconds.
35
Does the fraction 21/335 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 21/335 = 21/335. 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).
36
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.
37
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.
38
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.
39
Three warning lights flash every 17, 7, 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(17, 7, 12) = 1428 seconds.
40
Two buses leave a station together. One returns to the station every 21 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(21, 34) = 714 minutes.
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41
How many prime numbers are there between 44 and 69, inclusive?
Medium
ExplanationThe primes between 44 and 69 are [47, 53, 59, 61, 67], totaling 5.
42
Are 12 and 36 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(12, 36) = 12, so they are not relatively prime, since they share the common factor 12.
43
Is 413 a prime number?
Medium
Explanation413 is divisible by at least one number between 2 and 412, so it is not prime.
44
Does the fraction 28/355 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 28/355 = 28/355. 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).
45
What is the average (arithmetic mean) of the 7 consecutive integers from 71 to 77?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 74.
46
If two positive integers are 25 and 18, what is the product of their LCM and GCD?
Medium
ExplanationFor any two positive integers, LCM × GCD = product of the numbers = 25 × 18 = 450.
47
What is the smallest number that must be subtracted from 9514 so that the result is exactly divisible by 33?
Medium
Explanation9514 divided by 33 leaves remainder 10, so subtracting 10 makes it exactly divisible by 33.
48
What is the least common multiple (LCM) of 21 and 13?
Medium
ExplanationLCM(21, 13) = 273, the smallest number divisible by both 21 and 13.
49
The digits of 6475 sum to 22. Based on this, is 6475 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.
50
How many multiples of 7 are there between 39 and 129, inclusive?
Medium
ExplanationThe multiples of 7 in this range run from 42 to 126, giving 13 multiples.
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51
9704 is exactly divisible by which of the following numbers?
Medium
Explanation9704 ÷ 4 = 2426, a whole number, so 4 divides 9704 exactly.
52
28^2 − 20^2 is exactly divisible by which of the following?
Medium
Explanation28^2 − 20^2 = (28 − 20)(28 + 20) = 8 × 48, so it is always divisible by 8 (and by 48).
53
How many positive factors (divisors) does 126 have?
Medium
Explanation126 = 2^1 × 3^2 × 7^1. The number of divisors is the product of (exponent+1) for each prime factor = 12.
54
Three warning lights flash every 4, 14, 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(4, 14, 12) = 84 seconds.
55
What is the least common multiple (LCM) of 23 and 13?
Medium
ExplanationLCM(23, 13) = 299, the smallest number divisible by both 23 and 13.
56
What is the smallest positive integer that leaves a remainder of 7 when divided by 15 and a remainder of 3 when divided by 11?
Medium
ExplanationSince both remainders are exactly 8 less than their divisors, the answer is LCM(15, 11) − 8 = 165 − 8 = 157.
57
How many digits does 2^58 have?
Medium
ExplanationThe number of digits of 2^58 is floor(58 × log10(2)) + 1 = 18.
58
How many positive factors (divisors) does 24 have?
Medium
Explanation24 = 2^3 × 3^1. The number of divisors is the product of (exponent+1) for each prime factor = 8.
59
What is the sum of 6 consecutive integers starting from 48?
Medium
ExplanationThe integers are 48 through 53; their sum is 303.
60
How many multiples of 14 are there between 41 and 292, inclusive?
Medium
ExplanationThe multiples of 14 in this range run from 42 to 280, giving 18 multiples.
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61
4092 is exactly divisible by which of the following numbers?
Medium
Explanation4092 ÷ 3 = 1364, a whole number, so 3 divides 4092 exactly.
62
What is the greatest common divisor of 80, 90, and 130?
Medium
ExplanationThe largest integer dividing all of 80, 90, and 130 evenly is 10.
63
How many distinct prime factors does 115 have?
Medium
ExplanationThe prime factorization of 115 is 5^1 × 23^1, which includes 2 distinct prime(s).
64
Is 17361 divisible by 11?
Medium
ExplanationUsing the alternating digit-sum rule for 11, the result is -8, which is not a multiple of 11, so the answer is No.
65
What is the smallest positive number that must be added to 3604 so that the sum is exactly divisible by 16?
Medium
Explanation3604 divided by 16 leaves remainder 4, so adding 16 − 4 = 12 makes it exactly divisible by 16.
66
Is 63711 divisible by 11?
Medium
ExplanationUsing the alternating digit-sum rule for 11, the result is 10, which is not a multiple of 11, so the answer is No.
67
51^2 − 40^2 is exactly divisible by which of the following?
Medium
Explanation51^2 − 40^2 = (51 − 40)(51 + 40) = 11 × 91, so it is always divisible by 11 (and by 91).
68
What is the average (arithmetic mean) of the 7 consecutive integers from 29 to 35?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 32.
69
What is the sum of 3 consecutive integers starting from 12?
Medium
ExplanationThe integers are 12 through 14; their sum is 39.
70
How many positive factors (divisors) does 124 have?
Medium
Explanation124 = 2^2 × 31^1. The number of divisors is the product of (exponent+1) for each prime factor = 6.
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71
The digits of 4679 sum to 26. Based on this, is 4679 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 26, which is not divisible by 3.
72
What is the greatest common divisor of 105, 180, and 45?
Medium
ExplanationThe largest integer dividing all of 105, 180, and 45 evenly is 15.
73
Does the fraction 18/467 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 18/467 = 18/467. 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).
74
What is the units digit of 4^19?
Medium
ExplanationUnits digits of powers of 4 repeat in a cycle of length 2. Using 19 mod 2, the units digit is 4.
75
What is the smallest positive number that must be added to 1263 so that the sum is exactly divisible by 23?
Medium
Explanation1263 divided by 23 leaves remainder 21, so adding 23 − 21 = 2 makes it exactly divisible by 23.
76
What is the sum of all positive divisors of 133 (including 133 itself)?
Medium
ExplanationThe divisors of 133 are [1, 7, 19, 133], which sum to 160.
77
The digits of 8643 sum to 21. Based on this, is 8643 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 21, which is divisible by 3.
78
What is the sum of 3 consecutive integers starting from 3?
Medium
ExplanationThe integers are 3 through 5; their sum is 12.
79
What is the smallest number that must be subtracted from 5554 so that the result is exactly divisible by 14?
Medium
Explanation5554 divided by 14 leaves remainder 10, so subtracting 10 makes it exactly divisible by 14.
80
What is the greatest common divisor (GCD) of 270 and 162?
Medium
ExplanationThe largest integer dividing both 270 and 162 evenly is 54.
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81
The sum of an odd integer and an even integer is always
Medium
ExplanationThis follows directly from the basic parity rules of odd and even integers.
82
What is the smallest positive integer that leaves a remainder of 2 when divided by 5 and a remainder of 9 when divided by 12?
Medium
ExplanationSince both remainders are exactly 3 less than their divisors, the answer is LCM(5, 12) − 3 = 60 − 3 = 57.
83
What is the average (arithmetic mean) of the 5 consecutive integers from 44 to 48?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 46.
84
How many trailing zeros are in 53! (53 factorial)?
Medium
ExplanationTrailing zeros come from factors of 5 in 53!. Counting multiples of 5, 25, 125, ... gives 12 trailing zeros.
85
Are 75 and 61 relatively prime (i.e., is their greatest common divisor 1)?
Medium
ExplanationGCD(75, 61) = 1, so they are relatively prime.
86
What is the smallest positive number that must be added to 5410 so that the sum is exactly divisible by 37?
Medium
Explanation5410 divided by 37 leaves remainder 8, so adding 37 − 8 = 29 makes it exactly divisible by 37.
87
The digits of 489 sum to 21. Based on this, is 489 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 21, which is divisible by 3.
88
How many positive factors (divisors) does 141 have?
Medium
Explanation141 = 3^1 × 47^1. The number of divisors is the product of (exponent+1) for each prime factor = 4.
89
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.
90
How many distinct prime factors does 321 have?
Medium
ExplanationThe prime factorization of 321 is 3^1 × 107^1, which includes 2 distinct prime(s).
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91
What is the smallest positive number that must be added to 7082 so that the sum is exactly divisible by 43?
Medium
Explanation7082 divided by 43 leaves remainder 30, so adding 43 − 30 = 13 makes it exactly divisible by 43.
92
12^2 = 144. How many positive divisors does 144 have?
Medium
Explanation144 = 2^4 × 3^2, giving 15 divisors — always an odd count for a perfect square, since every divisor pairs with another except the square root.
93
Does the fraction 19/323 terminate as a decimal, or does it repeat?
Medium
ExplanationIn lowest terms, 19/323 = 1/17. 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).
94
What is the average (arithmetic mean) of the 5 consecutive integers from 60 to 64?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 62.
95
What is the least common multiple (LCM) of 38 and 13?
Medium
ExplanationLCM(38, 13) = 494, the smallest number divisible by both 38 and 13.
96
How many positive factors (divisors) does 109 have?
Medium
Explanation109 = 109^1. The number of divisors is the product of (exponent+1) for each prime factor = 2.
97
What is the smallest positive integer that leaves a remainder of 3 when divided by 4 and a remainder of 8 when divided by 9?
Medium
ExplanationSince both remainders are exactly 1 less than their divisors, the answer is LCM(4, 9) − 1 = 36 − 1 = 35.
98
What is the average (arithmetic mean) of the 5 consecutive integers from 2 to 6?
Medium
ExplanationFor consecutive integers, the average equals the middle value, which is 4.
99
The digits of 8793 sum to 27. Based on this, is 8793 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 27, which is divisible by 3.
100
What is the sum of 4 consecutive integers starting from 15?
Medium
ExplanationThe integers are 15 through 18; their sum is 66.