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saurabhmahajan
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Need to be moved to Data sufficiency forum.
Two approaches to show that stat (1) is acutally sufficient on it own: systematic examination of scenarios, or algebra:
systematic examination of scenarios: What could the units digit (and the resulting tens digit) be?
If the units digit is even, then the two digit number is automatically divisible by 2, making it a composite: 24, 36, 48 are all examples of composite two digit numbers where the tens digit is a factor of the units digit. So if the units digit is even we have composites - we just need to check whether can we find an example of a number that satisfies stat (1) but is not composite - Only within the odd community - those numbers with an odd units digit:
if the units digit is 1, then the tens must also be 1 to be a factor. You're basically left with 11, which can't be, because n must be greater than 20 (quest stem)
Units digit of 3: tens digit must be 1 or 3, since these are the only factors of 3. 15 is out (greater than 20), and 33 is composite (div by 3 and 11, so it's not prime).
units digit of 5: same as 3. 15 is out, and 55 is composite
units digit of 7: same as 3. 17 is out, and 77 is composite
unis digit of 9: allows 19, 39, 99 - 19 is smaller than 20, and the others are composite.
Thus, all ns that satisfy stat. (1) are composites, and the statement is actually sufficient on its own - no need for stat (2).
Algebra:
a two digit number XY (where X is the tens digit, Y is the units digit) can be written as
10X+Y
(for example, 53 is 5*10+3*1)
Now, stat. (1) basically tells you that X is a factor of Y: or in other words, Y is a multiple of X. Put another way, Y is equal to X*a, where a is some integer.
so our number 10X+y can be written as 10X + X*a. Such a number will definite be divisible by X: You can extract the common factor X from 10X + X*a to get X(10+a), so our two digit number is divisible by X. Now the question stem tells us that X is greater than 2 (the number is greater than 20, so tens digit is 2 and up), so whatever our two digit number is, it is not prime - it's divisible by X, which is neither 1 nor itself. Thus, our number is composite, and stat. (1) is sufficient again.
Two approaches to show that stat (1) is acutally sufficient on it own: systematic examination of scenarios, or algebra:
systematic examination of scenarios: What could the units digit (and the resulting tens digit) be?
If the units digit is even, then the two digit number is automatically divisible by 2, making it a composite: 24, 36, 48 are all examples of composite two digit numbers where the tens digit is a factor of the units digit. So if the units digit is even we have composites - we just need to check whether can we find an example of a number that satisfies stat (1) but is not composite - Only within the odd community - those numbers with an odd units digit:
if the units digit is 1, then the tens must also be 1 to be a factor. You're basically left with 11, which can't be, because n must be greater than 20 (quest stem)
Units digit of 3: tens digit must be 1 or 3, since these are the only factors of 3. 15 is out (greater than 20), and 33 is composite (div by 3 and 11, so it's not prime).
units digit of 5: same as 3. 15 is out, and 55 is composite
units digit of 7: same as 3. 17 is out, and 77 is composite
unis digit of 9: allows 19, 39, 99 - 19 is smaller than 20, and the others are composite.
Thus, all ns that satisfy stat. (1) are composites, and the statement is actually sufficient on its own - no need for stat (2).
Algebra:
a two digit number XY (where X is the tens digit, Y is the units digit) can be written as
10X+Y
(for example, 53 is 5*10+3*1)
Now, stat. (1) basically tells you that X is a factor of Y: or in other words, Y is a multiple of X. Put another way, Y is equal to X*a, where a is some integer.
so our number 10X+y can be written as 10X + X*a. Such a number will definite be divisible by X: You can extract the common factor X from 10X + X*a to get X(10+a), so our two digit number is divisible by X. Now the question stem tells us that X is greater than 2 (the number is greater than 20, so tens digit is 2 and up), so whatever our two digit number is, it is not prime - it's divisible by X, which is neither 1 nor itself. Thus, our number is composite, and stat. (1) is sufficient again.
