An integer n is called square-free if does not have a divisor of the form $k^{2}$ where $k \in {2, 3, \dots, n}$ . Find the number of square-free integers between $1$ and $120$ :

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Important Questions on Number System

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 159

How many positive integral solutions exist for:

ab + cd = a + b + c + d

, where

1 \leq a \leq b \leq c \leq d

?

MEDIUM

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 160

Ishaan Awasthi, a dyslexic boy, has some difficulties in reading but does not suffer from dyscalculia. Once his professor Mr. Nikumbh gave him a simple multiplication problem involving two distinct numbers. The numbers were exactly two-digit numbers, and none of the four digits was the same. He asked him to multiply both the numbers and find the answer. Due to inability in reading, Ishaan interchanged the positions of tens and unit digits mutually in each of the two numbers, and then he multiplied the resultant numbers. But the answer was still the same and correct. After realizing this amazing fact, now Nikumbh sir wants to know the number of pairs of two digit-numbers, which exists in our decimal number system.

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 1 - SNAP>CMAT>Q 164

What is the sum of all integers directly above $160$ ?