GMAT CLUB OLYMPICS: If x is a positive integer greater than 3 and x^4

Question

If x is a positive integer greater than 3 and  x^4 - 10x^2 + 9 = y , is y a multiple of 30 ?

(1) x is a prime number such that when x divided by 5 gives the remainder of 1
(2) x is a prime number such that when x divided by 6 gives the remainder of 5

Bunuel · Accepted Answer

Official Solution:

If  x  is a positive integer greater than 3 and  x^4 - 10x^2 + 9 = y , is  y  a multiple of 30 ? 
 
Work on the expression in the stem:  x^4 - 10x^2 + 9 = y ;
 
 x^4 - 9x^2 -x^2 + 9 = y ;
 
 x^2(x^2 - 9) -(x^2 - 9) = y ;
 
 (x^2 - 9)(x^2 -1) = y ;
 
 (x - 3)(x -1)(x+1)(x+3) = y ;
 
(1)  x  is a prime number such that when  x  divided by 5 gives the remainder of 1. 
 
First of all, since  x  is prime greater than 3, then  x  is odd. Now, if  x  is odd, then  (x - 3)(x -1)(x+1)(x+3)   is the product of four consecutive even integer, so even. Plus, out of four consecutive even integers, at least one will for sure be divisible by 3. Thus,  (x - 3)(x -1)(x+1)(x+3)   is divisible by  2*3=6 . 
 
Next, since  x = 5q + 1  ( x  divided by 5 gives the remainder of 1), then  (x - 3)(x -1)(x+1)(x+3) =(5q + 1 - 3)(5q + 1 -1)(5q + 1+1)(5q + 1+3) =(...)(5q)(...)(...)  is a multiple of 5. 
 
Therefore,  x^4 - 10x^2 + 9 = y  is a multiple of  6*5=30 . Sufficient.
 
(2)  x  is a prime number such that when  x  divided by 6 gives the remainder of 5
 
Test numbers:
 
If  x=5 , then  (x - 3)(x -1)(x+1)(x+3) = y  won't be a multiple of 5, so won't be a multiple of 30 either.
 
If  x=11 , then  (x - 3)(x -1)(x+1)(x+3) = 8*10*12*14 , which is a multiple of 2, 3, and 5, so a multiple of 30.
 
Two different answers. Not sufficient.

Answer: A

HarshaBujji · Answer

x can be {4,5,......}

x^4 - 10x^2 + 9  =>  (x^2 - 1) (x^2 - 9)  => (x-1)(x+1)(x-3)(x+3);

Let's use statement 1:   x is a prime number such that when x divided by 5 gives the remainder of 1

x = 5k + 1; Format; x is prime so k has to be even; Bcs x is a odd number

(x-1)(x+1)(x-3)(x+3) = 5k(5k+2)(5k-2)(5k+4) = 5k is multiple of 10;
 
We need to see if there is any 3 multiple;
x can be in any of the 2 forms 3n+1 or 3n+2; Can't be 3n as it is prime;

So x+1 or x-1 defiantly has to be a multiple of 3;

So (x-1)(x+1)(x-3)(x+3) is multiple of 30;

(2)  x is a prime number such that when x divided by 6 gives the remainder of 5 
x has to be in 6k+5 form;

So  (x-1)(x+1)(x-3)(x+3) = (6k+4)(6k+6)(6k+2)(6k+8) = 6K  format; K>1

=> x can be of 5n+1;5n+2;5n+3;5n+4 format x cannot be 5n as it is prime; where n>1

So (x-1)(x+1)(x-3)(x+3) any one of the above form will lead to a 5K form;   where K>1

Hence (x-1)(x+1)(x-3)(x+3) = 30k

But x =5; We get remainder as 5; But the (5-1)(5+1)(5-3)(5+3) not 30 multiple; Hence we cannot conclude it with statement 2 ;

IMO A

egmat · Answer

Key Step:  Factor the expression first.

x4 − 10x2 + 9 = (x2 − 1)(x2 − 9) =  (x−1)(x+1)(x−3)(x+3)

So y = (x−1)(x+1)(x−3)(x+3)

For y to be a multiple of 30, we need y divisible by 2, 3, and 5.

Critical Insight:  Since x is an odd prime > 3, all four factors are EVEN (divisibility by 2 ✓) and at least one factor is always divisible by 3 (✓). The only question is:  Is the product divisible by 5?

The product is divisible by 5  unless  x itself is a multiple of 5 (i.e., x = 5).

---

Statement 1:  x is prime with remainder 1 when divided by 5

→ x ≡ 1 (mod 5), which means (x − 1) is divisible by 5
→ Valid primes: 11, 31, 41, 61... (none equals 5)
→ Product is  always  divisible by 2 × 3 × 5 = 30

SUFFICIENT

---

Statement 2:  x is prime with remainder 5 when divided by 6

→ Valid primes: 5, 11, 17, 23, 29...

Case 1:  x = 5
y = (4)(6)(2)(8) = 384 = 27 × 3
 NOT divisible by 5 → NOT a multiple of 30

Case 2:  x = 11 
y = (10)(12)(8)(14) = 13,440
 Divisible by 30 ✓

Two valid cases → Two different answers

NOT SUFFICIENT

---
  Answer: A

GMAT CLUB OLYMPICS: If x is a positive integer greater than 3 and x^4

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

GMAT CLUB OLYMPICS: If x is a positive integer greater than 3 and x^4

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards