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State whether the given statement is true/false.

Let $a, b, c$ be positive integers such that $a$ divides $b 2$ , $b$ divides $c 2$ , and $c$ divides $a 2$ , then $a b c$ divides $(a + b + c) 7$ .

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True

False

answer is A.

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Detailed Solution

Given that

a

divides

b 2

b

divides

c 2

, and

c

divides

a 2

. So, the given condition is satisfied if

a = b = c

. Then, assume

a = b = c = k

.
Then, we get

(a + b + c) 7 = 37 × k 7

Also, the value of

a b c

a b c = k 3

.
Here, we see that

k 3 | (37 × k 7)

.
Rewriting the above statement in terms of a, b, c, and we get

a b c | (a + b + c) 7

Hence,

a b c

divides

(a + b + c) 7

.
So, the given statement is true.

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State whether the given statement is true/false.Let a,b,c be positive integers such that a divides b 2 , b divides c 2 , and c divides a 2 , then abc divides (a+b+c) 7 .

Detailed Solution

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State whether the given statement is true/false.

Let $a, b, c$ be positive integers such that $a$ divides $b 2$ , $b$ divides $c 2$ , and $c$ divides $a 2$ , then $a b c$ divides $(a + b + c) 7$ .