Playing with Numbers Test - 1

The number 88 can be written as

88 = 8 × 11, where 8 and 11 are co-primes.

It is known that if a number is divisible by two co-prime numbers, then it is divisible by their product too.

Therefore, any number is divisible by 88 if the number is divisible by both 8 and 11.

Consider the number 84920.

The number formed by the last three digits of this number is 920, which is a multiple of 8. Hence, the number is divisible by 8.

Sum of digits at odd places = 0 + 9 + 8 = 17

Sum of digits at even places = 2 + 4 = 6

Difference = 17 − 6 = 11, which is divisible by 11

Hence, the number is divisible by 11.

Hence, the number 84920 is divisible by 88.

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

A number is divisible by 10 if its last digit is 0.

Thus, only option D is correct. Since the last digit of 80 is 0, 80 is divisible by 10.

If a is a factor of b, then b is divisible by a. Here, 2 and 5 are factors of the required number. Thus, the number is divisible by both 2 and 5.

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

A number is divisible by 5 if its last digit is either 0 or 5.

Therefore, a number is divisible by 2 and 5 if its last digit is 0.

Among the options, only the number 30 has 0 as its last digit. Thus, 2 and 5 are both factors of 30.

A number is divisible by 2 if its units digit is even (0, 2, 4, 6, 8).

The units digit of 10,302 is 2. Therefore, 10,302 is divisible by 2.

A number is divisible by 3 if the sum of its digits is also divisible by 3.

For the number 10,302, sum of the digits = 1 + 0 + 3 + 0 + 2 = 6, which is divisible by 3.

Therefore, 10,302 is divisible by 3.

A number is divisible by 4 if the number formed from its last digits is 00 or is divisible by 4.

For the number 10,302, the number formed from its last 2 digits is 02, which is not divisible by 4. Thus, the number 10,302 is not divisible by 4.

A number is divisible by 5 if its units digit is 0 or 5.

The number 10,302 has the digit 2 at units place. Thus, it is not divisible by 5.

Thus, the given number is divisible by 2 and 3.

The given number is 11011110.

A number is divisible by 3 if the sum of its digits is divisible by 3.

Sum of the digits of the number 11011110 = 1 + 1 + 0 + 1 + 1 + 1 + 1 + 0 = 6, which is divisible by 3.

Therefore, the number 11011110 is divisible by 3.

It is known that if a number ends in 0, then it is divisible by 10.

Since the number 11011110 ends in 0, it is divisible by 10.

If, in a number, the difference between the sum of the digits at the odd places (starting from the right) and the sum of the digits at the even places (starting from the right) is either 0 or is divisible by 11, then the number is divisible by 11.

In the number 11011110,

Sum of the digits at the odd places = 1 + 1 + 1 + 0 = 3

Sum of the digits at the even places = 1 + 0 + 1 + 1 = 3

Difference = 3 – 3 = 0

Therefore, the number 11011110 is divisible by 11.

Thus, the number 11011110 is divisible by 3, 10, and 11.

The exact divisor of a number is called its factor.

48 = 1 × 48

= 2 × 24

= 3 ×16

= 4 × 12

= 6 × 8

Therefore, the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. Hence, the number of factors of 48 is 10.

Thus, the information in alternative A is correctly matched.

It is known that if two given numbers are divisible by a number, then their difference is also divisible by that number.

Therefore, in particular, if two numbers are divisible by 3, then their difference is also divisible by 3.

A number is even if its digit at units place is even, i.e. 0, 2, 4, 6, or 8.

Thus, both 28 and 42 are even.