how many 6 digit numbers of form (xyzzyx) where (y) is prime are possible which are divisible by 7
Answers
Answer:jord!!indiaan!!
Step-by-step explanation:
56 , 6 digit numbers of form (xyzzyx) where (y) is prime are possible which are divisible by 7
Step-by-step explanation:
6 digit numbers of form (xyzzyx) where (y) is prime
y = 2 , 3 , 5 , 7
Divisibility rule for 7
Take the digits of the number in reverse order, that is, from right to left, multiplying them successively by the digits 1, 3, 2, 6, 4, 5,
xyzzyx
= x * 1 + y * 3 + z * 2 + z * 6 + y * 4 + x * 5
= 6x + 7y + 8z
= 6x + x - x + 7y + 7z + z
= 7(x + y + z) + (z - x)
=> z - x = 0 or z - x = ±7
Case 1 : z - x = 0 => z = x
z & x can take 9 value from 1 to 9 ( x can not be zero)
y can take 4 Value (2 , 3 , 5 , 7)
so possible numbers = 4 * 9 = 36
Case 2 : z - x = ±7
Possible pairs of x & z
(1 , 8) ( 8 , 1) , (2 , 9 ) ( 9 , 2) , (7 , 0) ( x can not be zero)
y can take 4 Value (2 , 3 , 5 , 7)
so possible numbers = 5 * 4 = 20
36 + 20 = 56
56 , 6 digit numbers of form (xyzzyx) where (y) is prime are possible which are divisible by 7
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