Playing with Numbers Test - 6

Required interval is the least common factor of 3, 5, 7, 8 and 10.
LCM of 3, 5, 7, 8 and 10 = 840 seconds

Now, converting 840 seconds into minutes, we get: 840/60 = 14 minutes
Hence, they will again toll together after 14 minutes.

1. The sum of three odd numbers is even. False. Sum of three odd numbers is always odd.

2. Prime numbers do not have any factor. False. A prime number has two factors, one and the number itself.

3. The product of two even numbers is always even. True

4. All prime numbers are odd. False. 2 is an even prime number.

Statement-1 is true.

A number for which the sum of all its factors is equal to twice the number is called a perfect number.
Or in other words, a number for which the sum of its factors other than the number itself is equal to that number itself is called a perfect number.

For example,

Factors of 28: 1, 2, 4, 7, 14, 28
Sum of the factors other than 28 = 1 + 2 + 4 + 7 + 14 = 28

Statement-2 is true.

The product of three consecutive numbers will always be a multiple of 6.
For example: 7 x 8 x 9 = 504; and 504 is a multiple of 6.

(i) In case the two numbers are co-prime, their LCM will be the product of the numbers.
(ii) Let two consecutive numbers be 2n and 2n + 2. So, both have highest common factor 2, so HCF of two consecutive even numbers is 2.
(iii) If a number is divisible by 6, then it must also be divisible by 2 and 3.

If 821x is divisible by 8,
then, the only possible value of x is 6, when last three digits of 821x is divisible by 8.
So, x = 6

992y is divisible by 12,
9 + 9 + 2 + y = 20 + y is divisible by 12
y = 4

446z is divisible by 6,
4 + 4 + 6 + z = 14 + z is divisible by 6
z = 4
x + y + z = 6 + 4 + 4 = 14

46 is a multiple of 2.
121 is a multiple of 11.
49 is a multiple of 7.
81 is a multiple of 9.