A palindromic number is a positive integer one that reads the same for

Question

A palindromic number is a positive integer one that reads the same forwards and backwards. For example, 1221 is a palindromic number of four digits.

A positive integer is a perfect number if all of its positive integer factors less than the number sum up to the number itself. For example, 28 is a perfect number because all of its factors less than 28 (1, 2, 4, 7, and 14) sum up to 28.

N is a palindromic number of four digits such that the average of the four digits of N is 2.5. One of the digits of N is a prime number greater than 3. N can be expressed as a product of four positive odd numbers a, b, c, and d, each of which is greater than 1. P is defined as the average of a, b, c, and d. Q is defined as the perfect number closest to P.

Select for  (1)  the value of  P  and for  (2)  the value of  Q^2 / P .

Make only two selections, one in each column

Source : Manhattan

Rickooreoisb · Answer

Always remember 1001 = 7*11*13

Now number needs to have prime more than 3 - so only option are 5 and 7 
7 cannot be part because we need average to be 2.5 so sum to be 10

So number has to be 5005

Now 5005 = 5*7*11*13

Also, perfect numbers are 6, 28, 496 - Remember this!

So for 1 avergae of 5,7,11,13 is 9. and for 2 perfect number closest to 9 is 6. 
Q sqaure / 9 = 36/9=4

FrozenDragon · Answer

Rickooreoisb  Could you please edit the question, I have found multiple typos:

I think average should be 2.5 instead of 1.5

and instead of q2 to denote square, you can write it like this Q^2 for better understanding.

Thanks. Have a great day.

guddo · Answer

Rickooreoisb

Also, could you please cite the exact source of the question? That would be great.

Rickooreoisb · Answer

Gmat860sanskar   guddo  - thanks for highlighting updated the question

guddo · Answer

Rickooreoisb

Would you be kind enough to clarify the exact source of the question?

Rickooreoisb · Answer

I believe it is already there in the question, please check.

guddo · Answer

There are 2 Manhattans: Manhattan Prep (aka Manhattan GMAT) and Manhattan Review. Which one is it?

Rickooreoisb · Answer

MGMAT. Does it affect the learning form this question ? When we know it is Manhattan. Also going forward be specific in the query you have 
 
Let's focus on learning then having long thread on this please, thanks

guddo · Answer

Yes, it does make sense, and it  does  make a difference. We try to keep everything organized here. MGMAT (Manhattan Prep) and Manhattan Review are completely different companies, so giving the exact source matters.

Also, I was specific in my request. I asked for the  exact  source of the question. So in the future, please be precise when answering such queries.

egmat · Answer

Step 1: Find N (the 4-digit palindrome) 
A  4-digit palindrome  has the form  ABBA  (e.g.,  1221 ,  5005 ).

Sum of digits = A + B + B + A =  2A + 2B

Given: Average of digits =  2.5 
 →  Sum =  2.5 × 4 = 10 
 →   2A + 2B = 10 
 →   A + B = 5

Key constraint:  One digit must be a  prime number greater than 3 .
Single-digit primes >  3  are:  5, 7

Since A + B =  5 , and one digit must be  5  or  7 :
• If  7  is a digit → other digit would be negative (not possible)
•  Therefore , one digit must be  5

If A =  5 , then B =  0  → N =  5005  ✓
If B =  5 , then A =  0  →  Invalid (can't start with 0)

N = 5005

Step 2: Express N as product of 4 odd numbers > 1 
Factorize  5005 :
 5005 = 5 × 1001 = 5 × 7 × 11 × 13

So a, b, c, d =  5, 7, 11, 13  (all odd, all >  1  ✓)

Step 3: Calculate P 
P = Average of (a, b, c, d)
P = ( 5 + 7 + 11 + 13 ) /  4  =  36/4

P = 9

Step 4: Find Q (perfect number closest to P) 
 Perfect numbers : A number whose factors (excluding itself) sum to the number.
•  6  =  1 + 2 + 3  ✓
•  28  =  1 + 2 + 4 + 7 + 14  ✓

Distance from P =  9 :
• | 6 - 9 | =  3 
• | 28 - 9 | =  19

Q = 6   (closest to  9 )

Step 5: Calculate Q2/P 
Q2/P =  62 / 9  =  36/9  =  4

Answer: Part 1 = 9, Part 2 = 4

Adit_ · Answer

Spent way too much goofing around:
Average=2.5 so sum is 10
Palindrome and one number prime greater than 3
The number has to be 5 (7+7=14 would exceed 10)
Other number has to be 0 because of it cant have 0 at start so 5005 only possiblity

Factoring 5005=1001*5=5*7*11*13

Avg = 9
Perfect number closest to 9 = 6
Q^2/P = 4.

Answer: 9,4

A palindromic number is a positive integer one that reads the same for

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GMAT Two-part Analysis (TPA) Questions

A palindromic number is a positive integer one that reads the same for

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