Consider the number N = 774958P96Q
Q.1 If P = 2 and the number N is divisible by 3. then find the number of possible values of Q
Q.2 If N is divisible by 4, then find the number of possible ordered pairs (P. Q)
Q.3 If N is divisible by 8 and 9 both, then find the number of possible ordered pairs (PQ).
Answers
1. Therefore the number of possible values of Q is '3'.
2. Therefore the number of possible ordered pairs for ( P, Q ) is '30'.
3. Therefore the number of possible ordered pairs ( P, Q ) is '3'.
Given:
The number N = 774958P96Q
To Find:
1.) The number of possible values of Q, if P = 2 and the number N is divisible by 3.
2.) The number of possible ordered pairs (P, Q), if N is divisible by 4.
3.) The number of possible ordered pairs (P, Q), if N is divisible by 8 and 9 both.
Solution:
This question can be simply solved as shown below.
The given number N = 774958P96Q
1.) The number of possible values of Q, if P = 2 and the number N is divisible by 3.
If P = 2 then N = 774958296Q
Divisibility rule for '3':
The sum of the digits should be divisible by '3'.
⇒ Sum of digits in the given number N = 7+7+4+9+5+8+2+9+6+Q = 57 + Q
Now the sum of digits of the given number should be divisible by '3'.
⇒ ( 57 + Q ) / 3 = ( 19 + Q/3 )
Hence Q can be the multiples of '3' that is 3,6,9.
Therefore the number of possible values of Q is '3'.
2.) The number of possible ordered pairs (P, Q), if N is divisible by 4.
Divisibility rule for '4':
The last 2 digits in a number should be divisible by '4' if the number is divisible by '4'.
⇒ last 2 digits in the given number = 6Q
Now '6Q' should be divisible by '4' if the given number N should be divisible by '4'.
⇒ 60, 64, and 68 are the possible values that can be divisible by '4'. So the values of Q can be '0' or '4' or'8'.Hence the number of possible values for 'Q' is '3'.
⇒ P can take any number from 0 to 9. Hence the number of possible values for 'P' is '10'.
Hence the number of possible ordered pairs for ( P, Q ) = 10 × 3 = 30
Therefore the number of possible ordered pairs for ( P, Q ) is '30'.
3.) The number of possible ordered pairs (P, Q), if N is divisible by 8 and 9 both.
Divisibility rule for '8':
The last 3 digits in a number should be divisible by '8' if the number is divisible by '8'.
Divisibility rule for '9':
The sum of the digits should be divisible by '9'.
If a number is divisible by both '8' and '9' then both the conditions stated above should be satisfied.
Divisibility Test by '8',
⇒ Last 3 digits in the given number = 96Q
Now '96Q' should be divisible by '8' if the given number N should be divisible by '8'.
⇒ 960, and 968 are the possible values that can be divisible by '8'. So the values of Q can be '0' or '8'.Hence the number of possible values for 'Q' is '2'.
Divisibility Test by '9',
If Q = 0,
⇒ Sum of digits in the given number N = 7+7+4+9+5+8+P+9+6+0 = 55 + P
⇒ P = 8 is the only solution to get the sum divisible by '9'.
⇒ Hence one ordered pair ( P, Q ) = ( 8, 0 )
If Q = 8,
⇒ Sum of digits in the given number N = 7+7+4+9+5+8+P+9+6+8 = 63 + P
⇒ P = 0 or 9 are the solutions to get the sum divisible by '9'.
⇒ Hence one ordered pairs ( P, Q ) = ( 0, 8 ); ( 9, 8 )
Therefore the number of possible ordered pairs ( P, Q ) is '3'.
1. Therefore the number of possible values of Q is '3'.
2. Therefore the number of possible ordered pairs for ( P, Q ) is '30'.
3. Therefore the number of possible ordered pairs ( P, Q ) is '3'.
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