What is the Smallest? Trivia Quiz

Answer: 33

For a number to be a multiple of 3 and a multiple of 11, it must be a multiple of 33. 33 is the smallest number that fits.

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33...
Multiples of 11: 11, 22, 33...

Answer: 36

The sequence of triangular numbers begins with 1, 3, 6, 10, 15, 21, 28, 36...
The sequence of square numbers begins with 1, 4, 9, 16, 25, 36...

Other than 1, the first fit is the number 36. 1225 also fits, but 36 is smaller.

Answer: 21

Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21...
Multiples of 7: 7, 14, 21...

Interestingly, every eighth Fibonacci number is divisible by 7. The next is 987.

Answer: 252

Below 10, there are no multiples of 12.
Between 10 and 99, all palindromes are multiples of 11. The smallest positive integer that is both a multiple of 11 and a multiple of 12 is 132 (which is not a palindrome), so there aren't any in this range.
Between 100 and 199, palindromes have to end in 1, and numbers that end in 1 can't be multiples of 12.
Between 200 and 299, there could be one. Out of 202, 212, 222... ...292, there is only one multiple of 12, and that is 252 (12*21).

The next palindrome that is a multiple of 12 is 444 (12*37).

Answer: 189

63 is one less than 64 (4*4*4).
126 (63*2) is one more than 125 (5*5*5).
189 (63*3) is not one more or less than a cube, so it is the answer.

All cubes are either a multiple of 7, one less than a multiple of 7, or one more than a multiple of 7. The same is true for the number 9. This makes cubes adjacent to multiples of 63 much more common than the assumed expected chance of 2/63 by picking a random cubic number (in fact, 20 out of 63 cubic numbers are one more or one less than a multiple of 63).

Answer: 32768

The powers of 2 begin 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768...

7 is the last digit to appear in that sequence. It doesn't appear until 32768.

Two of the other options were powers, but not powers of 2. 1728 is 12*12*12, and 2187 is 3*3*3*3*3*3*3.

With any exponential sequence, approximately 30% of the terms will start with 1, and only about 5% will start with 9. This fact, combined with the fact that the last digit of any power of 2 must be even, makes the digits 7 and 9 first appear in the sequence later than other digits.

Answer: 144

Going through the list of squares:
1, 4, 9: It becomes the same number when reversed.
16, 25, 36, 49, 64, 81: These do not become squares when reversed.
100: Leading zeroes are not allowed.
121: It becomes the same number when reversed.
144: 441 (21*21) is a square! That means that 144 is the answer.

169 also fits (961 is 31*31), but 144 is smaller.

All numbers of this form begin and end with 1, 4, or 9. There are very few that have an even number of digits. Those that have odd numbers of digits also have their square roots reversed (for example, 12544=112*112, and 44521=211*211).

Answer: 5040

6=3!=3*2*1
24=4!=4*3*2
120=5!=6*5*4
720=6!=10*9*8

5040=7!, but 5040 is not the product of three consecutive numbers. 16*17*18=4896, while 17*18*19=5814.

If the question was reversed (a product of three consecutive numbers but not a factorial), the answer would be 60.

Answer: 106

The only even 2-digit palindromes are 22, 44, 66, and 88, and half of any of those numbers will still be a palindrome.

Numbers from 100 to 199 cannot be palindromes if they are even.

202 is a palindrome, but half of it is also a palindrome. 212 is a palindrome, and half of it is 106. Since we are looking for the number that is half of a palindrome, 106 is the answer.

Answer: It doesn't exist

A square of an even number will have an even digit in the ones place, since the number is even.

A square of an odd number will always have an even digit in the tens place and an odd digit in the ones place. If a square number ends in 1 or 9, its tens digit can be any even digit (0, 2, 4, 6, or 8), while if a square number ends in 5, its tens digit is always 2. (Square numbers can't end in 3 or 7.)

This means that either the ones or tens place (or both) will be even, and they cannot both be odd.