Playing with Numbers Test - 1

If a number is divisible by 4, it is always divisible by 2 as well because 4 is a multiple of 2. Every number in the times table of 4 is also in the times table of 2.
The number will also be divisible by 12 and 24 because 12 and 24 are both common multiples of 3 and 4.

Sum of odd places = 7 + 9 + 5 = 21
Sum of even places = 8 + a + 6 = 14 + a

If the number is divisible by 11, then either 21 - (14 + a) = 0 or 21 - (14 + a) = 11m, where m is any integer.
Therefore, a = 7

The numbers given are perfect square numbers.
Square numbers are numbers that we get by multiplying a number by itself once.

Square of 7 = 7 × 7 = 49
Square of 9 = 9 × 9 = 81

Highest three-digit number = 999
999 ÷ 17 = 58 Quotient; Remainder = 13

999 ÷ 29 = 34 Quotient; Remainder = 13
999 ÷ 39 = 25 Quotient; Remainder = 24

A composite number is a number that has exactly more than two factors.
Factors of 1 = 1

Factors of 4 = 1, 2 and 4
Factors of 8 = 1, 2, 4 and 8

Therefore, 1 is not a composite number because it has only 1 factor.

The first odd number is equal to the square of 1.
The sum of first two odd numbers is equal to the square of 2, and so on.

Therefore, the sum of first 13 odd numbers will be equal to the square of 13.
First 13 odd numbers go up to 25.

Therefore, 1 + 3 + 5 + ... + 25 will be equal to the square of 13.
13 × 13 = 169

The product of two prime numbers always has 4 factors.
The product itself and 1, and the two prime numbers.

15,540 is an even number, so it is divisible by 2.

A number is divisible by 3 if the sum of its digits is divisible by 3.
Sum of digits of 15,540 = 1+5+5+4+0= 15, which is divisible by 3, because 3×5 = 15
15,540 is divisible by 3 as well.

A number is divisible by 4 if the number formed by the last two digits of the number is divisible by 4.
Number formed by the last two digits of 15,540 = 40, which is divisible by 4, because 4×10 = 40
15,540 is divisible by 4 as well.

15,540 ends with '0', so it is divisible by 5.

A number is divisible by 6 if it is divisible by both 2 and 3.
15,540 is divisible by 6 as well.

To check if a number is divisible by 7, remove the last digit, double it, subtract it from the truncated original number and continue doing this until only one digit remains. If this is 0 or 7, then the original number is divisible by 7.

15,540 = 1554 - 2(0) = 1554
1554 = 155 - 2(4) = 147
147 = 14 - 2(7) = 0

Therefore, 15,540 is divisible by 7 as well.