GMAT Math: Number Theory
As the name suggests, most questions from number theory are about numbers, usually whole numbers or rational numbers. Number Theory questions are simple but not straightforward. Most GMAT problems should one way or the other involve number theory. Therefore some call it the heart of algebra and others the heart of mathematics. Unlike other topics, Number Theory has less rules and formulas. It is a subject with vast possibilities for application, which is why most test makers have a large scope of question models. Let's have a look at some problems so that we can understand the importance of basic Number Theory.
What is the remainder when 1088 * 1025 * 1070 * 1083 is divided by 31?
We need to know the basic rule of Number Theory (i.e. Rule of Remainders) to solve these types of problems.
Rule: When a number divides the product of two numbers, the remainder will be equal to the product of the remainders when the divisor divides the two numbers separately.
Assumptions before solving the problem play a key role here, so let's imagine that there is a number 'y' that can divide the product of P and Q.
Remainder of P/y is a
Remainder of Q/y is b
Remainder of (P*Q)/y is c
So c = a*b
Using the above rule,
The remainder when 31 divides 1088 is 3 .
The remainder when 31 divides 1025 is 2
The remainder when 31 divides 1070 is 16, and
The remainder when 31 divides 1083 is 29.
The net remainder is 3*2*16*29, which equals 2784
However, as the value of 3*2*16*29 is more than 31; the final remainder will be the remainder when 31 divides 21*24*27*30.
When 31 divides 2*16, the remainder is 1.
Similarly, when 31 divides 3*29, the remainder is 25.
The final remainder is 1*25=25.
The correct choice is (D) and the correct answer is 25
How many divisors will 1,800 have?
E. None of the above
(Questions of this type are interesting and easy to solve. We need to express the given number as a product of prime numbers.)
1,800 can be written as 18*100
18 and 100 are not primes, so we need to divide them until we get prime numbers.
18 can be written as 2*9. As 9 is not a prime so we can further write 18 as 2*3*3
Similarly, 100 can be written as 2*50, further as 2*2*25, and further again as 2*2*5*5
Combining the two above we can write 1,800 as 2*3*3*2*2*5*5
1,800 = 18*100
1,800 = 2*2*2*3*3*5*5
Here we have the prime factors 2, 3 and 5. We cannot divide them into further factors.
The prime factors will not play a key role here; rather the exponents of these prime factors being 3, 2, and 2 respectively play a key role.
We need to find the total number of integers, in other words find the total number of factors 1,800 has. That can be achieved by adding 1 to the exponents and multiplying them, as follows:
Number of factors = (3+1) * (2+1) * (2+1) = 36
The answer will therefore be (B)
How many trailing zeros will there be after the rightmost non-zero digit in the value of 30!?
Factorial questions are the most conceptual of Number Theory. Let's see the concept involved here:
We all know that 5*2 = 10, and any number multiplied by 10 or a power of 10 results in one or as many trailing zeroes as the power of 10. This means that 5 and 2 are the two basic prime numbers that give us a trailing zero. We now need to find the number of 2s and number of 5s in 30!. In other words, we need to find the numbers with 5 and 2 as factors, in order to find the number of trailing zeros.
30! = 30*29*28*..........2*1
Here we have 30, 25, 20, 15,10 and 5; these are the numbers where 5 is a factor. So can we say there are six 5s here? No, because we have 25 here, which has two 5s in it. So in total we have seven 5's.
We also have more than 7 even numbers in 30!, so we can judge the number of trailing zeros based on the number of 5s in 30!. The number of trailing zeroes is therefore 7
Hence the answer is (D)