ISI 2018

Number Theory Level 3

Let $a,b,c$ be natural numbers such that $a^{2}+b^{2}=c^{2},c-b=1$

Then State True or False for the following statements.

$a$ is even.

2. $b$ is divisible by 4.

3. $a^{b}+b^{a}$ is divisible by $c$ .

True,False,True False,True,True False,False,True False,True,False True,True,False True,True,True

2 solutions

Tom Engelsman
Dec 1, 2019

Let's examine the Pythagorean primitive $(a,b,c) = (3,4,5)$ :

(a) $a = 3$ is odd, hence FALSE,

(b) $b = 4$ is divisible by 4, hence TRUE,

Of course, there is not only the numbers that satisfy $a ^ { 2 } + b ^ { 2 } = c ^ { 2 }$ , and $c - b = 1$ is $( a, b, c ) = ( 3, 4, 5 )$ .

$( a, b, c ) = ( 5, 12, 13 ), ( 7, 24, 25 ), ( 9, 40, 41 ) \cdots$ satisfy the expressions above.

. . - 3 months, 2 weeks ago

. .
Feb 21, 2021

The simplest answer is $( a, b, c ) = ( 3, 4, 5 )$ .

$3$ is indivisible by 2, so it is $false$ .

$4$ is divisible by 4, so it is $true$ .

$3 ^ { 4 } + 4 ^ { 3 } = 81 + 64 = 145 \equiv 0 ( \mod 5 )$ , so it is $true$ .

So the answer is False, true, true.

ISI 2018

2 solutions

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