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If the seven digit number 74x29y6 is divisible by 72, then what will be the value of (2x + 3y)? (SSC Sub. Ins. 2018)
- a)21
- b)20
- c)19
- d)16
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
If the seven digit number 74x29y6 is divisible by 72, then what will b...
To determine the value of (2x + 3y), we need to find the values of x and y that make the seven-digit number 74x29y6 divisible by 72.
Divisibility Rule for 72:
A number is divisible by 72 if it is divisible by both 8 and 9.
Divisibility Rule for 8:
A number is divisible by 8 if the last three digits form a number divisible by 8.
Divisibility Rule for 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Finding the value of y:
Since the number is divisible by 8, the last three digits, 9y6, must be divisible by 8. The only possible values for y that make 9y6 divisible by 8 are 2 and 6. Therefore, y can be either 2 or 6.
Finding the value of x:
To find the value of x, we need to consider the divisibility rule for 9. The sum of the digits in the number 74x29y6 is 7 + 4 + x + 2 + 9 + y + 6 = 28 + x + y. This sum must be divisible by 9 since the number is divisible by 9.
For y = 2:
The sum of the digits is 28 + x + 2. To make this sum divisible by 9, x must be 7. Therefore, one possible value of x is 7.
For y = 6:
The sum of the digits is 28 + x + 6. To make this sum divisible by 9, x must be 1. Therefore, another possible value of x is 1.
Calculating (2x + 3y):
For y = 2 and x = 7, (2x + 3y) = (2*7 + 3*2) = 14 + 6 = 20.
For y = 6 and x = 1, (2x + 3y) = (2*1 + 3*6) = 2 + 18 = 20.
Therefore, the value of (2x + 3y) is 20, which corresponds to option (b).
Divisibility Rule for 72:
A number is divisible by 72 if it is divisible by both 8 and 9.
Divisibility Rule for 8:
A number is divisible by 8 if the last three digits form a number divisible by 8.
Divisibility Rule for 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Finding the value of y:
Since the number is divisible by 8, the last three digits, 9y6, must be divisible by 8. The only possible values for y that make 9y6 divisible by 8 are 2 and 6. Therefore, y can be either 2 or 6.
Finding the value of x:
To find the value of x, we need to consider the divisibility rule for 9. The sum of the digits in the number 74x29y6 is 7 + 4 + x + 2 + 9 + y + 6 = 28 + x + y. This sum must be divisible by 9 since the number is divisible by 9.
For y = 2:
The sum of the digits is 28 + x + 2. To make this sum divisible by 9, x must be 7. Therefore, one possible value of x is 7.
For y = 6:
The sum of the digits is 28 + x + 6. To make this sum divisible by 9, x must be 1. Therefore, another possible value of x is 1.
Calculating (2x + 3y):
For y = 2 and x = 7, (2x + 3y) = (2*7 + 3*2) = 14 + 6 = 20.
For y = 6 and x = 1, (2x + 3y) = (2*1 + 3*6) = 2 + 18 = 20.
Therefore, the value of (2x + 3y) is 20, which corresponds to option (b).
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Community Answer
If the seven digit number 74x29y6 is divisible by 72, then what will b...
Any number that is divisible by 72 must be divisible by 3, 4, 8, and 9.
Now, a number is divisible by 4 when a number formed by its last two digits of that number is divisible by 4. 'y6' is divisible by 4, for y = 1, 3, 5, 7, 9
Again a, number is divisible by 8, when a number formed by its last 3 digits is divisible by 8.
'9y6' is divisible by '8' for y = 3, 7
Now, for divisibility by 9, sum of its digits should be divisible by 9.
for y = 3, '7 + 4 + x + 2 + 9 + 3 + 6' = 31 + x
so, for x = 5, 36 is divisible by 9.
Now, is '7452936' which is divisible by '24' also, so, it is divisible '72'.
Now, (2x + 3y) = 2 × 5 + 3 × 3 = 19.
Now, a number is divisible by 4 when a number formed by its last two digits of that number is divisible by 4. 'y6' is divisible by 4, for y = 1, 3, 5, 7, 9
Again a, number is divisible by 8, when a number formed by its last 3 digits is divisible by 8.
'9y6' is divisible by '8' for y = 3, 7
Now, for divisibility by 9, sum of its digits should be divisible by 9.
for y = 3, '7 + 4 + x + 2 + 9 + 3 + 6' = 31 + x
so, for x = 5, 36 is divisible by 9.
Now, is '7452936' which is divisible by '24' also, so, it is divisible '72'.
Now, (2x + 3y) = 2 × 5 + 3 × 3 = 19.
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